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Brans–Dicke and General Scalar–Tensor Actions

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Scalar–tensor gravity promotes the effective strength of gravity to a field. In the Jordan description, matter follows one metric while a nonminimally coupled scalar changes both the metric equation and its own source. Eliminating the Ricci scalar with the metric trace reveals the central result: the matter trace, the scalar potential, and nonlinear coupling functions source one extra scalar degree of freedom. Radiation with vanishing trace is therefore different from nonrelativistic matter even before compact-object nonlinearities enter.

Required background. Einstein Equations supplies the metric variation, trace, and Bianchi identity used in the derivation.

Helpful background. Gravitational Actions and Boundary Terms supplies the boundary completion of the nonminimal curvature action; Formulations of General Relativity supplies the initial-value and constraint distinctions; Gravity Beyond General Relativity supplies the common theory record, frame, branch, and cutoff tests; Relativistic Matter and Compact Objects supplies the stress trace and stellar matter variables that source the scalar.

Jordan-frame field equations

Section titled “Jordan-frame field equations”

Use metric signature (−,+,+,+)(-,+,+,+) and let matter couple universally and minimally to gabg_{ab}. A broad two-derivative scalar–tensor action is

S=116πG∗∫Md4x−g[F(ϕ)R−Z(ϕ)∇aϕ∇aϕ−2U(ϕ)]+Sm[gab,ψ]+S∂M.\begin{aligned} S={}&\frac{1}{16\pi G_*}\int_M d^4x\sqrt{-g}\left[ F(\phi)R-Z(\phi)\nabla_a\phi\nabla^a\phi-2U(\phi) \right]\\ &+S_m[g_{ab},\psi]+S_{\partial M}. \end{aligned}

G∗G_* is a constant appearing in the action; it need not equal the Newtonian coupling measured on a particular background. FF, ZZ, and UU are fixed functions, F>0F\gt0 on the branch considered, and the boundary term is the one appropriate to the chosen scalar and induced-metric data. Varying the inverse metric gives

FGab=8πG∗Tab+Z(∇aϕ∇bϕ−12gab(∇ϕ)2)−Ugab+∇a∇bF−gab□F.\begin{aligned} F G_{ab}={}&8\pi G_*T_{ab} +Z\left(\nabla_a\phi\nabla_b\phi -\frac12g_{ab}(\nabla\phi)^2\right)-Ug_{ab}\\ &+\nabla_a\nabla_bF-g_{ab}\Box F. \end{aligned}

Varying the scalar independently gives

2Z□ϕ+Z,ϕ(∇ϕ)2+F,ϕR−2U,ϕ=0.2Z\Box\phi+Z_{,\phi}(\nabla\phi)^2 +F_{,\phi}R-2U_{,\phi}=0.

The second equation shows why setting the scalar to a constant is not automatically consistent. A constant ϕ0\phi_0 must make the curvature and potential terms balance on the selected matter background; it cannot be imposed after solving only the metric equation.

The same action and equations, with a notation in which FF is used as the scalar coordinate, appear in Clifton et al. 2012, §3.1.1, printed pp. 49–50, equations (95)–(99). Their conventions have been translated here to explicit G∗G_* and the site’s curvature and signature choices.

The trace exposes the scalar source

Section titled “The trace exposes the scalar source”

Taking the metric trace yields

FR=−8πG∗T+Z(∇ϕ)2+4U+3□F.FR=-8\pi G_*T+Z(\nabla\phi)^2+4U+3\Box F.

Use □F=F,ϕ□ϕ+F,ϕϕ(∇ϕ)2\Box F=F_{,\phi}\Box\phi+F_{,\phi\phi}(\nabla\phi)^2 in the scalar equation. After multiplying by FF and collecting terms,

(2ZF+3F,ϕ2)□ϕ=8πG∗F,ϕT+2FU,ϕ−4F,ϕU−(FZ,ϕ+F,ϕZ+3F,ϕF,ϕϕ)(∇ϕ)2.\begin{aligned} \left(2ZF+3F_{,\phi}^2\right)\Box\phi ={}&8\pi G_*F_{,\phi}T +2FU_{,\phi}-4F_{,\phi}U\\ &-\left( FZ_{,\phi}+F_{,\phi}Z+3F_{,\phi}F_{,\phi\phi} \right)(\nabla\phi)^2. \end{aligned}

This form separates three sources.

  • The matter trace enters with F,ϕF_{,\phi}. Conformal matter with T=0T=0 has no direct trace source, although it still gravitates through the metric and can be affected by a nontrivial scalar background.
  • 2FU,ϕ−4F,ϕU2FU_{,\phi}-4F_{,\phi}U selects constant-scalar vacua and determines the local scalar mass about them.
  • Derivatives of FF and ZZ produce nonlinear kinetic sourcing. They can become important even where a weakly gravitating body gives a small scalar.

The kinetic combination

K(ϕ)=2ZF+3F,ϕ2\mathcal K(\phi)=2ZF+3F_{,\phi}^2

also has a frame-invariant interpretation: after an invertible conformal map to an Einstein kinetic term, K/F2\mathcal K/F^2 controls the scalar field-space metric. A negative value on the F>0F\gt0 branch gives a wrong scalar kinetic sign. A zero value makes this canonical scalar map singular and requires a separate constraint analysis: the scalar can become auxiliary, as at the exceptional Brans–Dicke value ω=−3/2\omega=-3/2, or the perturbative description can become strongly coupled. A vanishing coefficient alone does not decide which possibility occurs.

Brans–Dicke form

Section titled “Brans–Dicke form”

Where F,ϕ≠0F_{,\phi}\ne0, use Φ=F(ϕ)\Phi=F(\phi) as the scalar coordinate and write

S=116πG∗∫d4x−g[ΦR−ω(Φ)Φ(∇Φ)2−2V(Φ)]+Sm+S∂M.S=\frac{1}{16\pi G_*}\int d^4x\sqrt{-g}\left[ \Phi R-\frac{\omega(\Phi)}{\Phi}(\nabla\Phi)^2-2V(\Phi) \right]+S_m+S_{\partial M}.

The metric equation is

ΦGab=8πG∗Tab+ωΦ(∇aΦ∇bΦ−12gab(∇Φ)2)+∇a∇bΦ−gab□Φ−Vgab,\begin{aligned} \Phi G_{ab}={}&8\pi G_*T_{ab} +\frac{\omega}{\Phi}\left( \nabla_a\Phi\nabla_b\Phi -\frac12g_{ab}(\nabla\Phi)^2\right)\\ &+\nabla_a\nabla_b\Phi-g_{ab}\Box\Phi-Vg_{ab}, \end{aligned}

and the scalar equation becomes

(2ω+3)□Φ=8πG∗T−ω,Φ(∇Φ)2+2ΦV,Φ−4V.(2\omega+3)\Box\Phi =8\pi G_*T -\omega_{,\Phi}(\nabla\Phi)^2 +2\Phi V_{,\Phi}-4V.

Constant ω\omega and V=0V=0 give the original Brans–Dicke model. The 2ω+32\omega+3 denominator is physical: it results from mixing between the metric trace and scalar kinetic term. The original construction and its trace-sourced scalar equation are given in Brans and Dicke 1961, pp. 925–927, equations (1)–(12).

For a constant vacuum Φ0\Phi_0, the background condition is

2Φ0V,Φ(Φ0)−4V(Φ0)=0.2\Phi_0V_{,\Phi}(\Phi_0)-4V(\Phi_0)=0.

Linearizing Φ=Φ0+δΦ\Phi=\Phi_0+\delta\Phi while neglecting background gradients gives

(□−mΦ2)δΦ=8πG∗2ω0+3 δT,(\Box-m_\Phi^2)\delta\Phi =\frac{8\pi G_*}{2\omega_0+3}\,\delta T,

with

mΦ2=2Φ0V,ΦΦ(Φ0)−2V,Φ(Φ0)2ω0+3.m_\Phi^2 =\frac{2\Phi_0V_{,\Phi\Phi}(\Phi_0)-2V_{,\Phi}(\Phi_0)} {2\omega_0+3}.

This formula assumes 2ω0+3≠02\omega_0+3\ne0 and a regular field coordinate. It distinguishes a long-range trace-coupled scalar from a short-range one without confusing either with the full nonlinear compact-object problem.

Which gravitational coupling is measured?

Section titled “Which gravitational coupling is measured?”

The coefficient G∗/Φ0G_*/\Phi_0 multiplies the tensor response on a nearly constant background, but a light scalar adds an attractive exchange between traceful sources. At separations short compared with its Compton wavelength, the Cavendish coupling in massless Brans–Dicke theory is

GN=G∗Φ02ω0+42ω0+3=G∗Φ0(1+12ω0+3).G_{\mathrm N} =\frac{G_*}{\Phi_0} \frac{2\omega_0+4}{2\omega_0+3} =\frac{G_*}{\Phi_0} \left(1+\frac{1}{2\omega_0+3}\right).

At distances long compared with a positive scalar Compton wavelength, the extra exchange is Yukawa suppressed and the response approaches G∗/Φ0G_*/\Phi_0. Thus “the gravitational constant” is background- and scale-qualified. Compact bodies add another layer: their masses can depend on the ambient scalar, producing body-dependent sensitivities and nonperturbative charges even when weakly gravitating matter couples weakly.

An independent normalization check comes from the Einstein description. The canonical scalar coupling satisfies α02=1/(2ω0+3)\alpha_0^2=1/(2\omega_0+3), while the conformal factor contributes A02=1/Φ0A_0^2=1/\Phi_0 in the present normalization. Exchange then gives GN=G∗A02(1+α02)G_{\mathrm N}=G_*A_0^2(1+\alpha_0^2), reproducing the Jordan-frame result. The next page tracks the accompanying masses, clocks, stress tensors, and dimensionless observables explicitly.

A controlled GR regime

Section titled “A controlled GR regime”

The formal statement ω→∞\omega\to\infty is insufficient on its own. A GR regime requires all of the following:

  1. Φ\Phi approaches a positive constant so that the matter metric and tensor coupling have a regular limit.
  2. The canonical scalar coupling to matter tends to zero, or its range becomes shorter than every scale used in the observable.
  3. Scalar initial and boundary data remain on the GR-connected branch rather than retaining a finite homogeneous wave or crossing a scalarized branch.
  4. Terms such as ω,Φ(∇Φ)2\omega_{,\Phi}(\nabla\Phi)^2 remain controlled; a divergent derivative of the coupling can spoil a naive large-ω\omega argument.
  5. The solution’s curvatures, gradients, frequencies, and nonlinearities stay below the EFT cutoff, and the dimensionless observable converges uniformly on the physical domain.

The conditions make room for two important possibilities: cosmological evolution can drive the ambient scalar toward weak coupling, while compact self-gravity can destabilize the weak-field branch and generate a finite charge. Those effects are developed later without importing empirical likelihoods or mutable bounds from Tests of Gravity.

Common pitfalls

Section titled “Common pitfalls”

Identify G∗G_* with the measured Newtonian coupling. A light scalar changes the force between traceful sources, so the measured coupling includes its exchange and depends on background and distance.

Set T=0T=0 and conclude the scalar is absent. The potential, background gradients, boundary data, and nonlinear kinetic terms can source the scalar even in vacuum.

Take ω→∞\omega\to\infty while holding arbitrary scalar data fixed. A decoupled source does not erase a pre-existing homogeneous scalar solution or guarantee that the selected branch approaches GR.

Exercises

Section titled “Exercises”

Starting from the Brans–Dicke-form metric equation, take its trace and combine it with the direct scalar variation. Recover the coefficient 2ω+32\omega+3 and the potential source 2ΦV,Φ−4V2\Phi V_{,\Phi}-4V. Then linearize about a constant vacuum and derive mΦ2m_\Phi^2.

Solution

The trace of the metric equation is

−ΦR=8πG∗T−ωΦ(∇Φ)2−3□Φ−4V,-\Phi R =8\pi G_*T -\frac{\omega}{\Phi}(\nabla\Phi)^2 -3\Box\Phi-4V,

or

ΦR=−8πG∗T+ωΦ(∇Φ)2+3□Φ+4V.\Phi R =-8\pi G_*T +\frac{\omega}{\Phi}(\nabla\Phi)^2 +3\Box\Phi+4V.

Direct scalar variation gives

R+2ωΦ□Φ+(ω,ΦΦ−ωΦ2)(∇Φ)2−2V,Φ=0.R+2\frac{\omega}{\Phi}\Box\Phi +\left( \frac{\omega_{,\Phi}}{\Phi}-\frac{\omega}{\Phi^2} \right)(\nabla\Phi)^2-2V_{,\Phi}=0.

Substituting the trace expression cancels the two terms proportional to ω(∇Φ)2/Φ\omega(\nabla\Phi)^2/\Phi and yields

(2ω+3)□Φ=8πG∗T−ω,Φ(∇Φ)2+2ΦV,Φ−4V.(2\omega+3)\Box\Phi =8\pi G_*T -\omega_{,\Phi}(\nabla\Phi)^2 +2\Phi V_{,\Phi}-4V.

For T=0T=0 and constant Φ0\Phi_0, the last two potential terms must cancel. Write Φ=Φ0+δΦ\Phi=\Phi_0+\delta\Phi and keep first order. The derivative of the potential source is

ddΦ(2ΦV,Φ−4V)Φ0=2Φ0V,ΦΦ(Φ0)−2V,Φ(Φ0).\frac{d}{d\Phi}\left(2\Phi V_{,\Phi}-4V\right)_{\Phi_0} =2\Phi_0V_{,\Phi\Phi}(\Phi_0)-2V_{,\Phi}(\Phi_0).

Moving this term to the left and dividing by 2ω0+32\omega_0+3 gives

(□−mΦ2)δΦ=0,mΦ2=2Φ0V,ΦΦ−2V,Φ2ω0+3.(\Box-m_\Phi^2)\delta\Phi=0, \qquad m_\Phi^2 =\frac{2\Phi_0V_{,\Phi\Phi}-2V_{,\Phi}}{2\omega_0+3}.

The sign of mΦ2m_\Phi^2 tests tachyonic stability only after the kinetic coefficient 2ω0+32\omega_0+3 is positive and the selected background solves both field equations.

References

Section titled “References”
  • Brans, Carl, and Robert H. Dicke (1961), “Mach’s Principle and a Relativistic Theory of Gravitation,” Physical Review 124, 925–935. DOI.
  • Clifton, Timothy, Pedro G. Ferreira, Antonio Padilla, and Constantinos Skordis (2012), “Modified Gravity and Cosmology,” Physics Reports 513, 1–189. DOI; Open PDF.