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3.2 The parametrized post-Newtonian formalism

Despite the possible existence of long-range gravitational fields in addition to the metric in various metric theories of gravity, the postulates of those theories demand that matter and non-gravitational fields be completely oblivious to them. The only gravitational field that enters the equations of motion is the metric g. The role of the other fields that a theory may contain can only be that of helping to generate the spacetime curvature associated with the metric. Matter may create these fields, and they plus the matter may generate the metric, but they cannot act back directly on the matter. Matter responds only to the metric.

Thus the metric and the equations of motion for matter become the primary entities for calculating observable effects, and all that distinguishes one metric theory from another is the particular way in which matter and possibly other gravitational fields generate the metric.

The comparison of metric theories of gravity with each other and with experiment becomes particularly simple when one takes the slow-motion, weak-field limit. This approximation, known as the post-Newtonian limit, is sufficiently accurate to encompass most solar-system tests that can be performed in the foreseeable future. It turns out that, in this limit, the spacetime metric g predicted by nearly every metric theory of gravity has the same structure. It can be written as an expansion about the Minkowski metric (η = diag(− 1,1,1,1) μν) in terms of dimensionless gravitational potentials of varying degrees of smallness. These potentials are constructed from the matter variables (see Box 2) in imitation of the Newtonian gravitational potential

∫ ′ ′ −1 3 ′ U (x,t) ≡ ρ(x ,t)|x − x | d x . (30 )
The “order of smallness” is determined according to the rules 2 U ∼ v ∼ Π ∼ p ∕ρ ∼ ε, vi ∼ |d∕dt|∕|d ∕dx| ∼ ε1∕2, and so on (we use units in which G = c = 1; see Box 2).

A consistent post-Newtonian limit requires determination of g00 correct through 𝒪 (ε2), g0i through 𝒪 (ε3∕2), and gij through 𝒪(ε) (for details see TEGP 4.1 [281Jump To The Next Citation Point]). The only way that one metric theory differs from another is in the numerical values of the coefficients that appear in front of the metric potentials. The parametrized post-Newtonian (PPN) formalism inserts parameters in place of these coefficients, parameters whose values depend on the theory under study. In the current version of the PPN formalism, summarized in Box 2, ten parameters are used, chosen in such a manner that they measure or indicate general properties of metric theories of gravity (see Table 2). Under reasonable assumptions about the kinds of potentials that can be present at post-Newtonian order (basically only Poisson-like potentials), one finds that ten PPN parameters exhaust the possibilities.

Table 2: The PPN Parameters and their significance (note that α3 has been shown twice to indicate that it is a measure of two effects).


What it measures relative to GR

Value in GR
Value in semiconservative theories
Value in fully conservative theories


How much space-curvature produced by unit rest mass?

1 γ γ


How much “nonlinearity” in the superposition law for gravity?

1 β β


Preferred-location effects?

0 ξ ξ


Preferred-frame effects?

0 α1 0


0 α2 0


0 0 0


Violation of conservation

0 0 0


of total momentum?

0 0 0


0 0 0


0 0 0


0 0 0

The parameters γ and β are the usual Eddington–Robertson–Schiff parameters used to describe the “classical” tests of GR, and are in some sense the most important; they are the only non-zero parameters in GR and scalar-tensor gravity. The parameter ξ is non-zero in any theory of gravity that predicts preferred-location effects such as a galaxy-induced anisotropy in the local gravitational constant GL (also called “Whitehead” effects); α1, α2, α3 measure whether or not the theory predicts post-Newtonian preferred-frame effects; α3, ζ1, ζ2, ζ3, ζ4 measure whether or not the theory predicts violations of global conservation laws for total momentum. In Table 2 we show the values these parameters take

  1. in GR,
  2. in any theory of gravity that possesses conservation laws for total momentum, called “semi-conservative” (any theory that is based on an invariant action principle is semi-conservative), and
  3. in any theory that in addition possesses six global conservation laws for angular momentum, called “fully conservative” (such theories automatically predict no post-Newtonian preferred-frame effects).

Semi-conservative theories have five free PPN parameters (γ, β, ξ, α 1, α 2) while fully conservative theories have three (γ, β, ξ).

The PPN formalism was pioneered by Kenneth Nordtvedt [197], who studied the post-Newtonian metric of a system of gravitating point masses, extending earlier work by Eddington, Robertson and Schiff (TEGP 4.2 [281Jump To The Next Citation Point]). Will [274] generalized the framework to perfect fluids. A general and unified version of the PPN formalism was developed by Will and Nordtvedt. The canonical version, with conventions altered to be more in accord with standard textbooks such as [189Jump To The Next Citation Point], is discussed in detail in TEGP 4 [281Jump To The Next Citation Point]. Other versions of the PPN formalism have been developed to deal with point masses with charge, fluid with anisotropic stresses, bodies with strong internal gravity, and post-post-Newtonian effects (TEGP 4.2, 14.2 [281Jump To The Next Citation Point]).


Box 2. The Parametrized Post-Newtonian formalism


Coordinate system:
The framework uses a nearly globally Lorentz coordinate system in which the coordinates are (tx1x2x3). Three-dimensional, Euclidean vector notation is used throughout. All coordinate arbitrariness (“gauge freedom”) has been removed by specialization of the coordinates to the standard PPN gauge (TEGP 4.2 [281Jump To The Next Citation Point]). Units are chosen so that G = c = 1, where G is the physically measured Newtonian constant far from the solar system.
Matter variables:
PPN parameters:
γ, β, ξ, α1, α2, α3, ζ1, ζ2, ζ3, ζ4.
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 − (α1 − α2 − α3)w2U − α2wiwjUij i 3 + (2α3 − α1)w Vi + 𝒪 (ε ), 1 1 1 g0i = −--(4γ + 3 + α1 − α2 + ζ1 − 2 ξ)Vi −-(1 + α2 − ζ1 + 2ξ )Wi − --(α1 − 2α2)wiU 2 2 2 − α2wjUij + 𝒪(ε5∕2), 2 gij = (1 + 2γU )δij + 𝒪 (ε ).
Metric potentials:
∫ ---ρ′--- 3 ′ U = |x − x ′| d x , ∫ ρ ′(x − x′)i(x − x ′)j Uij = ------------′3-----d3x ′, |x − x | ∫ ρ ′ρ′′(x − x ′) ( x ′ − x′′ x − x′′) ΦW = ------------ ⋅ --------− --------- d3x′d3x′′, |x − x′|3 |x − x′′| |x ′ − x ′′| ∫ ′ ′ ′ 2 𝒜 = ρ-[v-⋅ (x-−-x-)]-d3x ′, |x − x′|3 ∫ --ρ′v′2-- 3 ′ Φ1 = |x − x ′| d x , ∫ ρ′U ′ 3 ′ Φ2 = |x-−-x-′| d x , ∫ ρ′Π ′ Φ3 = -------′d3x ′, |x − x | ∫ p′ Φ4 = --------d3x ′, |x − x ′| ∫ ′′ Vi = --ρ-vi--d3x ′, |x − x ′| ∫ ′ ′ ′ ′ W = ρ-[v-⋅ (x-−-x-)](x −-x)i d3x′. i |x − x ′|3
Stress–energy tensor (perfect fluid):
T 00 = ρ (1 + Π + v2 + 2U ), ( ) 0i i 2 -p T = ρv 1 + Π + v + 2U + ρ , ( ) T ij = ρvivj 1 + Π + v2 + 2U + -p + pδij(1 − 2γU ). ρ
Equations of motion:

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