This section is a recap of the Parameterised Post-Newtonian (PPN) formalism that is widely used by both theoretical and observational gravitational physicists. The idea here is to create a construction that encompasses a wide array of different gravitational theories, and that contains parameters that can be constrained by observations in a rea- sonably straightforward fashion. In this way labour can be saved on both the theoretical and observational ends of the spectrum: Observers can apply their results to constrain a wide array of theories without having to trawl through the details of the individual theories themselves, and theorists can straightforwardly constrain their new theories by comparing to the already established bounds on the PPN parameters without having to re-calculate individual gravitational phenomena. To date, this approach has been highly successful, and in the following sections of this report we will often refer to it. We will therefore outline here how the PPN formalism proceeds. For a more detailed explanation of the principles and consequences of this formalism the reader is referred to [1274]. 2.5.1. Parameterised post-Newtonian formalism
The PPN formalism is a perturbative treatment of weak-field gravity, and therefore requires a small parameter to expand in. For this purpose an “order of smallness” is defined by
U ∼ v2
∼ Pρ ∼ Π ∼ O(2),
where U is the Newtonian potential, v is the 3-velocity of a fluid element, P is the pressure of the fluid, ρ is its rest-mass density and Π is the ratio of energy density to rest-mass density. Time derivatives are also taken to have an order of smallness associated with them, relative to spatial derivatives:
|∂/∂t|
|∂/∂x| ∼ O(1).
Here we have chosen to set c = 1. The PPN formalism now proceeds as an expansion in this order of smallness.
For time-like particles coupled to the metric only the equations of motion show that the level of approximation required to recover the Newtonian limit is g00 to O(2), with
no other knowledge of other metric components beyond the background level being nec- essary. The post-Newtonian limit for time-like particles, however, requires a knowledge of
g00 to O(4)
g0i to O(3)
gij to O(2).
Latin letters here are used to denote spatial indices. To obtain the Newtonian limit of null particles we only need to know the metric to background order: Light follows straight lines, to Newtonian accuracy. The post-Newtonian limit of null particles requires a knowledge of g00 and gij both to O(2).
The way in which the PPN formalism then proceeds is as follows. First one identifies the different fields in the theory. All dynamical fields should then be perturbed from
their expected background values, and the perturbations assigned an appropriate order of smallness each. For theories containing a metric the appropriate expansion is usually
g00 = −1 + h(2)00 + h (4)
00 + O(6) (61)
g0i = h(3)0i + O(5) (62)
gij = δij+ h(2)ij + O(4), (63)
where superscripts in brackets denote the order of smallness of the term. If, for example, the theory contains an additional scalar field, then the usual expansion for this quantity is
φ = φ0+ ϕ(2)+ ϕ(4)+ O(6), (64)
where φ0 is the constant background value of φ. Additional vector and tensor gravita-
tional fields can be specified in a corresponding way.
The energy-momentum tensor in the PPN formalism is then taken to be that of a perfect fluid. To the relevant order, the components of this tensor are given by
T00= ρ(1 + Π + v2− h00) + O(6) (65)
T0i=−ρvi+ O(5) (66)
Tij= ρvivj+ P δij+ O(6). (67)
Taking these expressions, the field equations for the theory in question, and substituting in the perturbed expressions for the dynamical fields in the theory, as prescribed above, the field equations can then be solved for order by order in the smallness parameter.
The first step in such calculations is usually to solve for h(2)00. With this solution in hand, one then proceeds to solve for h(2)ij and h(3)0i simultaneously, and finally h(4)00 can be solved for. If additional fields exist, beyond the metric, then these quantities must also be solved for to increasing order of smallness as the calculation proceeds. In finding h(2)ij , h(3)0i and h
(4)
00 one needs to specify a gauge. After such a specification one still, of course,
has the freedom to make additional gauge transformations of the form xµ → xµ+ ξµ,
where ξµ is O(2) or smaller. This freedom should be used at the end of the process to
transform the metric that has been obtained into the “standard post-Newtonian gauge”. This is a gauge in which the spatial part of the metric is diagonal, and terms containing time derivatives are removed. Once this has been done then one is in possession of the PPN limit of the theory in question.
We have so far outlined the procedure that one needs to follow in order to gain the appropriate form of the metric that couples to matter fields in the weak-field limit. Once done, the result can then be compared to the ‘PPN test metric’ below:
g00 = −1 + 2GU − 2βG2U2− 2ξG2ΦW + (2γ + 2 + α3+ β1− 2ξ)GΦ1 +2(1 + 3γ− 2β + β2+ ξ)G2Φ2+ 2(1 + β3)GΦ3− (β1− 2ξ)GA +2(3γ + 3β4− 2ξ)GΦ4 g0i = − 1 2(3 + 4γ + α1− α2+ β1− 2ξ)GVi− 1 2(1 + α2− β1+ 2ξ)GWi gij = (1 + 2γGU )δij. 33
Here β, γ, ξ, β1, β2, β3, β4, α1, α2 and α3 are the ‘post-Newtonian parameters’, U is
the Newtonian gravitational potential that solves the Newtonian Poisson equation, and ΦW, Φ1, Φ2, Φ3, Φ4, A, Vi and Wi are the ‘post-Newtonian gravitational potentials’
(the precise form of these potentials is given in [1274]). The particular combination of parameters before each of these potentials is chosen here so that they have particular physical significance, once gravitational phenomena have been computed.
2.5.2. Parameterised post-Newtonian constraints
Comparison of the weak field metric of a particular theory with the PPN test metric above allows one to read off values for the PPN parameters β, γ, ξ, β1, β2, β3, β4, α1, α2
and α3 for the theory in question. The test metric has been constructed to include the
type of potentials that often appear when one modifies gravity10. The great utility of
the PPN formalism is that observers can take the PPN test metric above and constrain the parameters without having a particular theory in mind. These constraints can then be applied directly to a large number of gravitational theories, without having to work out how complicated gravitational phenomena work in each theory individually.
In General Relativity we have that β = γ = 1 and ξ = β1 = β2= β3= β4 = α1 =
α2= α3= 0. Other theories will predict other values for these parameters, and we will
discuss these on a case by case basis in the sections that follow. Observationally, one can use the gravitational phenomena discussed in Section 1 to impose the constraints that follow.
As already discussed, observations that involve only null geodesics are sensitive to the Newtonian part of the metric, g00(2), and the term g
(2)
ij only. These two terms involve
the PPN parameter γ only. We can now use constraints on the bending of light by the Sun to get a constraint on γ. Using the PPN test metric the predicted bending of light that one should observe is [1274]
θ = 2(1 + γ)m
r =
(1 + γ)
2 θGR, (68)
where m is the mass of the Sun, r is its radius, and θGR is the general relativistic
prediction. Using the observed value of θ given in Section 1 then gives [1131]
γ− 1 = (−1.7 ± 4.5) × 10−4, (69)
which is consistent with the general relativistic value of γ = 1. Similarly, we can use the PPN test metric to find that the Shapiro time delay effect is given by [1274]
∆t = (1 + γ)
2 ∆tGR, (70)
where subscript GR again means the value of this quantity as predicted by General Relativity. Taking the observed value of ∆t given in Section 1 then gives the even tighter constraint [147]
γ− 1 = (2.1 ± 2.3) × 10−5, (71)
10It is not, however, an exhaustive collection of all possible potentials, and in some theories it is
again consistent with γ = 1. It can now be clearly seen that the bending of light by the Sun, and the Shapiro time delay effect do, in fact, constrain the same aspect of space-time geometry. They can therefore be considered as complimentary to each other.
If we now consider observations of gravitational phenomena that involve time-like geodesics then we are able to observe, potentially, all of the post-Newtonian potentials in the PPN test metric. This becomes clear from the expression for perihelion precession, which now becomes
∆ω = 6πM p 1 3(2 + 2γ− β) + 1 6(2α1− α2+ α3+ 2β2) µ M + J2 r2 2M p , where M is the total mass of the two bodies involved, µ is their reduced mass, and p is the semi-latus rectum of the orbit. The affect of modifying the geometry can be seen here to be degenerate with the effect due to the solar quadrupole moment, J2. Once
the value of this quantity is known, however, then one is able to gain constraints on the above combination of β, γ, α1, α2, α3 and β2. This can be done for any or all of
the observations of the perihelion precession of Mercury given in Section 1, and if we take the value of γ to be that given by Eq. (71), as well as11 α
1 ∼ α2 ∼ α3 ∼ β2 ∼ 0
and a reasonable value of J2 ∼ 10−7, then this gives constraints on β of the order
β− 1 ∼ O(10−3) or O(10−4). However, as already noted, these constraints are somewhat
sensitive to a number of assumptions about the orbits of the other planets, as well as the solar quadrupole moment.
The Nordtvedt effect is similarly an observation of time-like geodesics. In this case it is convenient to define the ‘Nordtvedt parameter’
ηN ≡ 4β − γ − 3 − 10 3 ξ− α1+ 2 3α2− 2 3β1− 1 3β2, (72)
which is not to be confused with the equivalence principle violation parameter η defined in Equation (15). The observations of Williams, Turyshev and Boggs [1277] then give the constraint ηN = (4.4± 4.5) × 10−4, which, if we again take γ to be given by observations
of the Shapiro time delay effect with all other PPN parameters being zero, gives us
β− 1 = (1.2 ± 1.1) × 10−4, (73)
which is a much cleaner constraint on β than those which can be derived from observations of the perihelion precession of Mercury.
In ‘conservative’ theories of gravity it is usually only the PPN parameters β and γ (and sometimes ξ) that vary from their general relativistic values. These quantities are often interpreted as the degree of non-linearity in the gravitational theory, and the amount of spatial curvature per unit mass that is produced, respectively. The other parameters ξ, αi and βi are usually interpreted as corresponding to preferred location
effects, preferred frame effect and the violation of conserved quantities. When considering theories in which such effects are expected to be absent it is therefore usual to assume that these parameters are all zero, and to search instead for constraints on β and γ.
Of course, one can subject the ξ, αi and βi parameters to observational scrutiny in a
number of ways. The table below gives a selection of the tightest constraints currently available:
11These values will be given some justification shortly. 35
Parameter Limit Source
ξ 10−3 Ocean tides [1274]
α1 10−3 Lunar laser ranging [928]
α2 4× 10−7 Alignment of Sun’s spin axis with ecliptic [987]
α3 4× 10−20 Pulsar acceleration [1176]
Further constraints and discussion on the βiparameters can be found in [1274]. For more
details of the observations leading to these constraints on ξ and αithe reader is referred
to the source material cited above and [1274].
The constraints on the PPN parameters that we have discussed above are all, to date, in reasonably good agreement with General Relativity, and it is likely that future observations of, for example, the ‘double pulsar’ [840, 745] will tighten these constraints even further in coming years. This excellent concordance of numerous different physical phenomena means that one must reconcile any alterations to General Relativity with observations in weak field systems that appear to be narrowing down on a general rel- ativistic description. As we will describe in the sections that follow, this places tight constraints on a variety of different modified theories of gravity: It must be the case that any alternative theories that we consider should reproduce General Relativity in the appropriate weak field limit, or at least something very close to it.
There are a number of mechanisms that have been considered in the literature that allow for a general relativistic weak field limit even in theories that are, in general, very different from General Relativity. These include the Vainshtein mechanism [1241] which occurs when large derivative interactions are present, the Chameleon mechanism of theo- ries with non-minimal coupling to scalar fields [689], as well as the attractor mechanism of Damour and Nordtvedt [356]. These different approaches allow, potentially, for theories that deviate considerably from General Relativity to exist without disturbing gravita- tional physics in the solar system to a large extent. They are thought to be successful in a number of different environments, and have sometimes been applied to situations that are quite different to the ones in which they were originally conceived.
As well as successful reproductions of general relativistic behaviour, however, there have also been a number cases found in the literature of theories that produce weak field gravity that is surprisingly inconsistent with the predictions of General Relativity. Per- haps the most famous of these is the van Dam-Veltman-Zakharov (vDVZ) discontinuity that was originally found in the context of Pauli-Fierz gravity [1243, 1296] (a theory with one dynamical metric, and one non-dynamical a priori specified metric). Here the gravi- ton acquires a mass through the introduction of terms into the gravitational Lagrangian that, in the weak field limit, look like mass terms for the perturbations hµν around
Minkowski space, i.e. like m2hµνh
µν. Naively one might then expect in the limit m→ 0,
when the graviton becomes massless, that the zero mass theory of General Relativity should be recovered. This is, however, not the case. Instead one finds from the study of linear perturbations around Minkowski space that γ→ 1/2, which can be seen from the constraints above to be in strong disagreement with a number of different observations, including light bending and time delay effects. The general relativistic limit in this case is therefore a singular one, and any finite but non-zero graviton mass, no matter how small, appears to give results that are incompatible with observations. Similar results
have also been found in some theories of gravity constructed from more general functions of the Ricci curvature than the Einstein-Hilbert action [294], and are expected in other theories as well. In these cases one must either abandon the theory as being incompatible with observations, or show that the treatment being applied is unsatisfactory because, for example, one of the mechanisms discussed previously should be applied.
Issues such as those just discussed can make the study of weak field gravity in modified theories a more complicated subject than it is in General Relativity. One must be careful to make sure that the treatments being applied are justifiable, that the limits of the theory take the expected form (rather than being singular), and that non-linear mechanisms and non-perturbative effects are being fully taken into account. How this should be done for specific modified theories of gravity will be the subject of subsequent sections. In some cases it is still an active area of research.