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3.3 Competing theories of gravity

One of the important applications of the PPN formalism is the comparison and classification of alternative metric theories of gravity. The population of viable theories has fluctuated over the years as new effects and tests have been discovered, largely through the use of the PPN framework, which eliminated many theories thought previously to be viable. The theory population has also fluctuated as new, potentially viable theories have been invented.

In this review, we shall focus on GR, the general class of scalar-tensor modifications of it, of which the Jordan–Fierz–Brans–Dicke theory (Brans–Dicke, for short) is the classic example, and vector-tensor theories. The reasons are several-fold:

3.3.1 General relativity

The metric g is the sole dynamical field, and the theory contains no arbitrary functions or parameters, apart from the value of the Newtonian coupling constant G, which is measurable in laboratory experiments. Throughout this article, we ignore the cosmological constant ΛC. We do this despite recent evidence, from supernova data, of an accelerating universe, which would indicate either a non-zero cosmological constant or a dynamical “dark energy” contributing about 70 percent of the critical density. Although ΛC has significance for quantum field theory, quantum gravity, and cosmology, on the scale of the solar-system or of stellar systems its effects are negligible, for the values of ΛC inferred from supernova observations.

The field equations of GR are derivable from an invariant action principle δI = 0, where

∫ I = (16πG )− 1 R(− g)1∕2d4x + Im(ψm, gμν), (31 )
where R is the Ricci scalar, and I m is the matter action, which depends on matter fields ψ m universally coupled to the metric g. By varying the action with respect to gμν, we obtain the field equations
G ≡ R − 1g R = 8πGT , (32 ) μν μν 2 μν μν
where Tμν is the matter energy-momentum tensor. General covariance of the matter action implies the equations of motion μν T ;ν = 0; varying Im with respect to ψm yields the matter field equations of the Standard Model. By virtue of the absence of prior-geometric elements, the equations of motion are also a consequence of the field equations via the Bianchi identities μν G ;ν = 0.

The general procedure for deriving the post-Newtonian limit of metric theories is spelled out in TEGP 5.1 [281Jump To The Next Citation Point], and is described in detail for GR in TEGP 5.2 [281Jump To The Next Citation Point]. The PPN parameter values are listed in Table 3.

Table 3: Metric theories and their PPN parameter values (α3 = ζi = 0 for all cases). The parameters γ ′, β′, α′1, and α ′2 denote complicated functions of u and of the arbitrary constants. Here Λ is not the cosmological constant ΛC, but is defined by Equation (37View Equation).
Arbitrary functions
or constants
Cosmic matching
PPN parameters
γ β ξ α1 α2
General relativity none none 1 1 0 0 0
Brans–Dicke ωBD φ0 1 + ωBD -------- 2 + ωBD 1 0 0 0
General A (ϕ), V (ϕ) ϕ0 1 + ω ------ 2 + ω 1 + Λ 0 0 0
Unconstrained ω,c1,c2,c3,c4 u ′ γ ′ β 0 ′ α 1 ′ α2
Einstein-Æther c1,c2,c3,c4 none 1 1 0 α ′ 1 α′ 2
Rosen’s bimetric none c0,c1 1 1 0 0 c0 c1 − 1

3.3.2 Scalar-tensor theories

These theories contain the metric g, a scalar field ϕ, a potential function V (ϕ ), and a coupling function A (ϕ) (generalizations to more than one scalar field have also been carried out [73Jump To The Next Citation Point]). For some purposes, the action is conveniently written in a non-metric representation, sometimes denoted the “Einstein frame”, in which the gravitational action looks exactly like that of GR:

∫ [ ] &tidle; − 1 &tidle; μν 1∕2 4 ( 2 ) I = (16πG ) R − 2&tidle;g ∂ μϕ∂ νϕ − V (ϕ) (− &tidle;g) d x + Im ψm, A (ϕ )&tidle;gμν , (33 )
where &tidle; μν &tidle; R ≡ g&tidle; Rμν is the Ricci scalar of the “Einstein” metric g&tidle;μν. (Apart from the scalar potential term V(ϕ ), this corresponds to Equation (28View Equation) with &tidle;G (ϕ) ≡ (4πG )−1, U (ϕ ) ≡ 1, and M&tidle; (ϕ) ∝ A (ϕ ).) This representation is a “non-metric” one because the matter fields ψm couple to a combination of ϕ and &tidle;gμν. Despite appearances, however, it is a metric theory, because it can be put into a metric representation by identifying the “physical metric”
gμν ≡ A2 (ϕ)&tidle;gμν. (34 )
The action can then be rewritten in the metric form
−1∫ [ −1 μν 2 ] 1∕2 4 I = (16 πG ) φR − φ ω (φ )g ∂μφ∂νφ − φ V (− g) d x + Im(ψm, gμν), (35 )
−2 φ ≡ A(ϕ ) , − 2 3 + 2ω (φ) ≡ α(ϕ ) , (36 ) d(lnA (ϕ)) α (ϕ ) ≡ ----dϕ----.
The Einstein frame is useful for discussing general characteristics of such theories, and for some cosmological applications, while the metric representation is most useful for calculating observable effects. The field equations, post-Newtonian limit and PPN parameters are discussed in TEGP 5.3 [281Jump To The Next Citation Point], and the values of the PPN parameters are listed in Table 3.

The parameters that enter the post-Newtonian limit are

[ ] dω − 2 −1 ω ≡ ω (φ0), Λ ≡ d-φ(3 + 2ω ) (4 + 2ω ) , (37 ) φ0
where φ0 is the value of φ today far from the system being studied, as determined by appropriate cosmological boundary conditions. In Brans–Dicke theory (ω (φ ) ≡ ω = const. BD), the larger the value of ωBD, the smaller the effects of the scalar field, and in the limit ωBD → ∞ (α0 → 0), the theory becomes indistinguishable from GR in all its predictions. In more general theories, the function ω (φ ) could have the property that, at the present epoch, and in weak-field situations, the value of the scalar field φ0 is such that ω is very large and Λ is very small (theory almost identical to GR today), but that for past or future values of φ, or in strong-field regions such as the interiors of neutron stars, ω and Λ could take on values that would lead to significant differences from GR. It is useful to point out that all versions of scalar-tensor gravity predict that γ ≤ 1 (see Table 3).

Damour and Esposito-Farèse [73] have adopted an alternative parametrization of scalar-tensor theories, in which one expands lnA (ϕ ) about a cosmological background field value ϕ0:

ln A (ϕ) = α0(ϕ − ϕ0 ) + 1β0(ϕ − ϕ0 )2 + ... (38 ) 2
A precisely linear coupling function produces Brans–Dicke theory, with α20 = 1∕(2ωBD + 3), or 1∕(2 + ω ) = 2α2∕ (1 + α2 ) BD 0 0. The function ln A(ϕ ) acts as a potential for the scalar field ϕ within matter, and, if β0 > 0, then during cosmological evolution, the scalar field naturally evolves toward the minimum of the potential, i.e. toward α0 ≈ 0, ω → ∞, or toward a theory close to, though not precisely GR [8081]. Estimates of the expected relic deviations from GR today in such theories depend on the cosmological model, but range from 10–5 to a few times 10–7 for |γ − 1|.

Negative values of β0 correspond to a “locally unstable” scalar potential (the overall theory is still stable in the sense of having no tachyons or ghosts). In this case, objects such as neutron stars can experience a “spontaneous scalarization”, whereby the interior values of ϕ can take on values very different from the exterior values, through non-linear interactions between strong gravity and the scalar field, dramatically affecting the stars’ internal structure and leading to strong violations of SEP. On the other hand, in the case β0 < 0, one must confront that fact that, with an unstable ϕ potential, cosmological evolution would presumably drive the system away from the peak where α0 ≈ 0, toward parameter values that could be excluded by solar system experiments.

Scalar fields coupled to gravity or matter are also ubiquitous in particle-physics-inspired models of unification, such as string theory [254176858283]. In some models, the coupling to matter may lead to violations of EEP, which could be tested or bounded by the experiments described in Section 2.1. In many models the scalar field could be massive; if the Compton wavelength is of macroscopic scale, its effects are those of a “fifth force”. Only if the theory can be cast as a metric theory with a scalar field of infinite range or of range long compared to the scale of the system in question (solar system) can the PPN framework be strictly applied. If the mass of the scalar field is sufficiently large that its range is microscopic, then, on solar-system scales, the scalar field is suppressed, and the theory is essentially equivalent to general relativity.

3.3.3 Vector-tensor theories

These theories contain the metric g and a dynamical, typically timelike, four-vector field uμ. In some models, the four-vector is unconstrained, while in others, called Einstein-Æther theories it is constrained to be timelike with unit norm. The most general action for such theories that is quadratic in derivatives of the vector is given by

∫ − 1 [ μ μν α β μ ] 1∕2 4 I = (16πG ) (1 + ωu μu )R − K αβ∇ μu ∇ νu + λ (u μu + 1) (− g) d x + Im(ψm, gμν), (39 )
μν μν μ ν μ ν μ ν K αβ = c1g gαβ + c2δαδβ + c3δβδα − c4u u gαβ. (40 )
The coefficients c i are arbitrary. In the unconstrained theories, λ ≡ 0 and ω is arbitrary. In the constrained theories, λ is a Lagrange multiplier, and by virtue of the constraint μ uμu = − 1, the factor μ ωu μu in front of the Ricci scalar can be absorbed into a rescaling of G; equivalently, in the constrained theories, we can set ω = 0. Note that the possible term uμu νRμν can be shown under integration by parts to be equivalent to a linear combination of the terms involving c2 and c 3.

Unconstrained theories were studied during the 1970s as “straw-man” alternatives to GR. In addition to having up to four arbitrary parameters, they also left the magnitude of the vector field arbitrary, since it satisfies a linear homogenous vacuum field equation of the form ℒu μ = 0 (c4 = 0 in all such cases studied). Indeed, this latter fact was one of most serious defects of these theories. Each theory studied corresponds to a special case of the action (39View Equation), all with λ ≡ 0:

General vector-tensor theory; ω, τ, ε, η (see TEGP 5.4 [281Jump To The Next Citation Point])
The gravitational Lagrangian for this class of theories had the form R + ωuμu μR + ηu μuνR μν − εF μνF μν + τ ∇μu ν∇ μuν, where F μν = ∇ μuν − ∇ νuμ, corresponding to the values c = 2 ε − τ 1, c = − η 2, c + c + c = − τ 1 2 3, c = 0 4. In these theories γ, β, α1, and α2 are complicated functions of the parameters and of u2 = − uμuμ, while the rest vanish.

Will–Nordtvedt theory (see [290])
This is the special case c1 = − 1, c2 = c3 = c4 = 0. In this theory, the PPN parameters are given by γ = β = 1, α2 = u2∕(1 + u2∕2), and zero for the rest.

Hellings–Nordtvedt theory; ω (see [128])
This is the special case c1 = 2, c2 = 2ω, c1 + c2 + c3 = 0 = c4. Here γ, β, α1 and α2 are complicated functions of the parameters and of u2, while the rest vanish.

The Einstein-Æther theories were motivated in part by a desire to explore possibilities for violations of Lorentz invariance in gravity, in parallel with similar studies in matter interactions, such as the SME. The general class of theories was analyzed by Jacobson and collaborators [13718313899113Jump To The Next Citation Point], motivated in part by [156].

Analyzing the post-Newtonian limit, they were able to infer values of the PPN parameters γ and β as follows [113]:

γ = 1, (41 ) β = 1, (42 ) ξ = α = ζ = ζ = ζ = ζ , (43 ) 3 12 2 3 4 α = − -8(c3 +-c1c4)-, (44 ) 1 2c1 − c21 + c23 2 2 2 α2 = (2c13-−-c14) − 12c3c13-+-2c2c14(1 −-2c14)-+-(c1 −-c3)(4-−-6c13 +-7c14), (45 ) c123(2 − c14) (2 − c14)(2c1 − c21 + c23)
where c = c + c + c 123 1 2 3, c = c − c 13 1 3, c = c − c 14 1 4, subject to the constraints c ⁄= 0 123, c ⁄= 2 14, 2 2 2c1 − c1 + c3 ⁄= 0.

By requiring that gravitational wave modes have real (as opposed to imaginary) frequencies, one can impose the bounds c1∕(c1 + c4) ≥ 0 and (c1 + c2 + c3)∕(c1 + c4) ≥ 0. Considerations of positivity of energy impose the constraints c1 > 0, c1 + c4 > 0 and c1 + c2 + c3 > 0.

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