Conformal Frames and Observable Quantities

Frame invariance is an operational statement: two sets of field variables give the same dimensionless result when the matter metric, sources, clocks, masses, boundary data, and observable are transformed together. In scalar–tensor gravity this rule is especially concrete. A Jordan description gives matter a fixed mass and a nonminimally coupled metric; an Einstein description gives the metric a canonical kinetic term and matter a scalar-dependent mass. Either description is complete, but mixing their measuring conventions is not.

Required background. Einstein Equations supplies the curvature transformation and tensor response used below.

Helpful background. Gravitational Actions and Boundary Terms supplies the surface contribution transformed with the nonminimal action; Formulations of General Relativity supplies the initial and asymptotic data that must map between descriptions; Gravity Beyond General Relativity supplies the general invertibility and observable criteria; Relativistic Matter and Compact Objects supplies the material clocks, stress tensors, and compact-body masses whose transformations are physical checks.

From Jordan variables to canonical tensor variables

Start from the Jordan action on a connected branch with $F(\phi)\gt0$,

$$S_J=\frac{1}{16\pi G_*}\int d^4x\sqrt{-g_J} \left[F(\phi)R_J-Z(\phi)(\nabla_J\phi)^2-2U(\phi)\right] +S_m[g^J_{ab},\psi]+S^J_{\partial M}.$$

Choose the asymptotic normalization $F_0=F(\phi_0)$ and define

$$g^E_{ab}=\frac{F(\phi)}{F_0}g^J_{ab}, \qquad A^2(\phi)=\frac{F_0}{F(\phi)}, \qquad g^J_{ab}=A^2g^E_{ab}.$$

The normalization $A(\phi_0)=1$ makes the two asymptotic unit systems coincide; another constant choice changes dimensionful labels but not dimensionless observables. In four dimensions,

$$R_J=A^{-2}\left[ R_E-6\Box_E\ln A-6(\nabla_E\ln A)^2 \right].$$

After combining the divergence with the transformed boundary action and introducing a canonical scalar $\varphi$, the theory takes the form

$$\begin{aligned} S_E={}&\frac{1}{16\pi G_E}\int d^4x\sqrt{-g_E} \left[R_E-2(\nabla_E\varphi)^2-4V_E(\varphi)\right]\\ &+S_m[A^2(\varphi)g^E_{ab},\psi]+S^E_{\partial M}. \end{aligned}$$

The constants and functions in this expression are fixed by the Jordan action, rather than introduced independently:

$$G_E=\frac{G_*}{F_0},\qquad \left(\frac{d\varphi}{d\phi}\right)^2 =\frac{Z}{2F}+\frac34\left(\frac{F_{,\phi}}{F}\right)^2, \qquad V_E=\frac{F_0U}{2F^2}.$$

The scalar $\varphi$ is dimensionless with the conventional kinetic factor $-2/(16\pi G_E)$; a mass-dimension-one canonical scalar is $\chi=\varphi/\sqrt{4\pi G_E}$. This dictionary fixes the otherwise easy-to-miss factors of two in scalar charges and potentials.

The coefficient convention for $\varphi$ is chosen so that

$$\alpha(\varphi)=\frac{d\ln A}{d\varphi}$$

is the dimensionless scalar coupling. The physical matter metric is still $\widetilde g_{ab}=A^2g^E_{ab}=g^J_{ab}$. The Einstein metric is a convenient tensor variable, not a declaration that freely falling matter follows its timelike geodesics. Damour and Esposito-Farèse make the same separation between the Einstein kinetic metric and physical metric in Damour and Esposito-Farèse 1996, §I, pp. 1475–1477, equations (1.1)–(1.3).

Stress tensors and material motion

Define stress tensors by variation with respect to each inverse metric. At fixed $\varphi$,

$$T^E_{ab}=A^2T^J_{ab}, \qquad T_E^{ab}=A^6T_J^{ab}, \qquad T_E=A^4T_J.$$

Jordan-frame minimal coupling gives $\nabla^J_aT_J^{ab}=0$. In Einstein variables the same identity becomes

$$\nabla^E_aT_E^{ab} =\alpha(\varphi)T_E\nabla_E^b\varphi.$$

The matter stress tensor alone is not conserved with the Einstein connection because it exchanges energy and momentum with the scalar; the combined Einstein-frame scalar-plus-matter stress is conserved. This is a translation of one theory, not a violation of diffeomorphism invariance. The transformed conservation law and conformal geometry are derived in Faraoni and Nadeau 2007, §2, pp. 4–6, equations (2.1)–(2.13).

For a point particle of fixed Jordan mass $m_J$,

$$S_{\mathrm{pp}} =-m_J\int ds_J =-\int m_E(\varphi)\,ds_E, \qquad m_E=A(\varphi)m_J.$$

Its Einstein-frame acceleration is

$$u_E^b\nabla^E_bu_E^a =-\left(g_E^{ab}+u_E^au_E^b\right) \nabla^E_b\ln A.$$

The right-hand side is the same physical motion that the Jordan description calls geodesic. It arises because the Einstein-frame mass standard varies. Running masses, units, and the timelike equation are treated explicitly in Faraoni and Nadeau 2007, §§3.1–3.2, pp. 7–12, equations (3.1)–(3.22).

A dictionary of measured quantities

Let a subscript $J$ or $E$ denote the proper-time and mass unit used in that description at the same event. The transformation rules are

Quantity	Jordan variables	Einstein variables	Invariant comparison
Proper interval	$ds_J$	$ds_E=A^{-1}ds_J$	Number of realized clock periods along the worldline
Material rest mass	$m_J$	$m_E=A m_J$	Ratio to a material mass standard with the same coupling
Photon frequency	$\omega_J=-k_au_J^a$	$\omega_E=A\omega_J$	$\omega/m_{\mathrm{clock}}$ at emission or reception
Matter trace	$T_J$	$T_E=A^4T_J$	Scalar source after its field normalization and volume element are included
Effective point-particle mass	$m_A^J(\phi_\infty)$	$m_A^E=A(\varphi_\infty)m_A^J$	Dimensionless orbital phase or binding-energy comparison with the mass and scalar coupling consistently mapped
Weak-field force	Tensor plus nonminimal scalar response	Canonical tensor plus explicit scalar exchange	$G_EA_0^2(1+\alpha_0^2)$ for two weak, traceful bodies when the scalar is effectively massless; here $A_0=1$ and $G_E=G_*/F_0$

The photon covector $k_a=\nabla_a\vartheta$ is the gradient of an invariant phase. Since $d\tau_J=A\,d\tau_E$, the four-velocities satisfy $u_J^a=A^{-1}u_E^a$ and hence $\omega_E=A\omega_J$. An atomic transition has $m_E=A m_J$, so the locally measured ratio obeys

$$\frac{\omega_E}{m_{E,\mathrm{clock}}} =\frac{A\omega_J}{A m_{J,\mathrm{clock}}} =\frac{\omega_J}{m_{J,\mathrm{clock}}}.$$

The same logic applies at emission and reception, producing a frame-invariant redshift when both clocks and the null phase are transformed. A statement that the photon frequency changed while the atomic standard did not has held one frame’s unit fixed inside the other frame’s equations.

Compact bodies and asymptotic normalization

Strong self-gravity makes a body’s mass a function of the ambient scalar. In Einstein variables its scalar charge is summarized by

$$\alpha_A(\varphi_\infty) =\frac{\partial\ln m_A^E(\varphi_\infty)} {\partial\varphi_\infty}$$

at fixed baryon number or another declared material invariant. The derivative contains both the universal factor $A$ and the scalar dependence of the binding energy. Consequently a weak-field coupling $\alpha_0$ cannot simply be substituted for a neutron star’s charge. The asymptotic conformal normalization, mass definition, and quantity held fixed must accompany the number.

The coefficient of a scalar’s $1/r$ tail, and its ratio to mass, also depend on the scalar normalization; they are not independently invariant numbers. The table’s mass transformation follows from the effective point-particle action and must not be applied blindly to an ADM surface integral in a nonminimally coupled description. Predictions such as a dimensionless binding energy or orbital phase agree when the complete body model, surface-charge definition, scalar normalization, and asymptotic units are mapped. Empirical likelihoods and numerical bounds on those predictions remain the responsibility of Tests of Gravity.

Equivalence domain and failure modes

The healthy canonical description used here requires a smooth one-to-one map on the entire branch, $F\gt0$, a positive finite scalar field-space metric, and mapped initial, boundary, horizon, and asymptotic data. Positivity is a health condition, not a general requirement for a change of variables: a regular negative scalar kinetic coefficient can also be transformed, but the ghost remains in the new frame. The canonical map can fail at $F=0$, at a turning point of the scalar redefinition, or when finite regular data map to a singular configuration. A disformal extension

$$\bar g_{ab}=C(\phi)g_{ab} +D(\phi)\nabla_a\phi\nabla_b\phi$$

also requires $C\gt0$ and $C-2DX\gt0$ in the site convention $X=-(\nabla\phi)^2/2$, together with a nonsingular field-space Jacobian. The opposite sign for $X$ in Bettoni and Liberati is translated in Bettoni and Liberati 2013, §II.B, pp. 3–5, equations (6)–(14).

Quantum equivalence adds the functional measure, regulator, counterterms, and possible anomalies. A classical conformal calculation does not settle those questions. Nor does choosing $g_E$ as a new minimally coupled matter metric: that is a different theory, not another frame of the Jordan model.

Common pitfalls

Compare dimensional masses between frames. A mass is measured against a material standard that transforms with it. Compare a dimensionless ratio.

Call $T_E$ nonconservation a physical loss of energy. The scalar carries the compensating exchange; the total Einstein-frame stress obeys the Bianchi identity.

Use the weak-field coupling for a compact body’s charge. Binding energy and branch selection make $\alpha_A$ body-dependent and potentially nonperturbative.

Exercises

A photon emitted by one atomic transition is received by an identical atomic clock. Starting from $g^J_{ab}=A^2g^E_{ab}$, show that the local frequency ratio at each endpoint and therefore the measured redshift are the same in both frames.

Solution

The scalar phase $\vartheta$ is unchanged by a field redefinition, so its covector $k_a=\nabla_a\vartheta$ is common to both descriptions. Proper times obey $d\tau_J=A\,d\tau_E$, hence

$$u_J^a=\frac{dx^a}{d\tau_J}=A^{-1}u_E^a.$$

The locally measured photon frequencies therefore satisfy

$$\omega_J=-k_au_J^a=A^{-1}(-k_au_E^a)=A^{-1}\omega_E.$$

The point-particle or atomic matter action gives $m_{E,\mathrm{at}}=A m_{J,\mathrm{at}}$. At either endpoint $P$,

$$\left.\frac{\omega_E}{m_{E,\mathrm{at}}}\right|_P =\left.\frac{A\omega_J}{A m_{J,\mathrm{at}}}\right|_P =\left.\frac{\omega_J}{m_{J,\mathrm{at}}}\right|_P.$$

The observed redshift is the ratio of the emitted and received dimensionless frequencies, so both endpoint factors agree between frames even when $A(\phi)$ differs at emission and reception. Comparing $\omega_E$ with a fictitiously fixed Jordan atomic mass would leave the matter action untransformed and describe a different experiment.

References

• Bettoni, Dario, and Stefano Liberati (2013), “Disformal Invariance of Second Order Scalar-Tensor Theories: Framing the Horndeski Action,” Physical Review D 88, 084020. DOI; Open PDF.
• Damour, Thibault, and Gilles Esposito-Farèse (1996), “Tensor-Scalar Gravity and Binary-Pulsar Experiments,” Physical Review D 54, 1474–1491. DOI; Open PDF.
• Faraoni, Valerio, and Shahn Nadeau (2007), “The (Pseudo)issue of the Conformal Frame Revisited,” Physical Review D 75, 023501. DOI; Open PDF.