Jordan, Einstein, and Other Field Frames

Jordan and Einstein frames are different field-coordinate systems only when the transformation is smooth and invertible and every part of the theory is carried along. Transforming the gravitational action while leaving matter, sources, units, or boundary data behind defines a different physical model. The invariant comparison is therefore between dimensionless observables built from the metric seen by matter, not between individual dimensionful quantities written in one set of variables.

Required background. Einstein Equations supplies the GR normalization and Bianchi identity; Gravitational Actions and Boundary Terms supplies the surface terms that must transform with the bulk action; Variational and Symplectic Geometry supplies the criterion that an invertible field map preserve the reduced phase space. The complete theory data being transformed were defined on Actions, Fields, and Symmetries.

Helpful background. Formulations of General Relativity supplies the constraint and initial-data language needed to compare the same solution branch in two field frames.

Conformal maps and the measuring metric

Let two Lorentzian metrics be related in four dimensions by

$$\widetilde g_{ab}=\Omega^2(\phi)g_{ab}, \qquad \Omega\gt0.$$

With all derivatives on the right taken using $g_{ab}$,

$$\sqrt{-\widetilde g}=\Omega^4\sqrt{-g}, \qquad \widetilde R =\Omega^{-2}\left[ R-6\Box\ln\Omega-6(\nabla\ln\Omega)^2 \right].$$

The $\Box\ln\Omega$ term is a bulk divergence only after its induced boundary term has been retained. The transformation is therefore a map of the complete action, including boundary completion, rather than a license to drop the second-derivative term without comment. The conformal formulas and transformed matter conservation law are derived in Faraoni and Nadeau 2007, §2, pp. 4–6, equations (2.1)–(2.13).

Consider a Jordan-frame scalar–tensor action

$$\begin{aligned} S_J={}&\int_M d^4x\sqrt{-g_J}\left[ \frac12F(\phi)R_J -\frac12Z(\phi)(\nabla_J\phi)^2-U(\phi) \right]\\ &+S_m[g^J_{ab},\psi]+S_{\partial M}. \end{aligned}$$

The name Jordan frame means here that matter is minimally and universally coupled to $g^J_{ab}$; it is not determined merely by the appearance of $F(\phi)R_J$. On a connected domain where $F\gt0$, define

$$g^E_{ab}=\frac{F(\phi)}{M_{\mathrm{Pl}}^2}g^J_{ab}, \qquad M_{\mathrm{Pl}}^{-2}=8\pi G.$$

After the conformal transformation and one integration by parts, with its surface contribution combined into the transformed boundary action,

$$\begin{aligned} S_E={}&\int_M d^4x\sqrt{-g_E}\left[ \frac{M_{\mathrm{Pl}}^2}{2}R_E -\frac12K(\phi)(\nabla_E\phi)^2 -V_E(\phi) \right]\\ &+S_m[A^2(\phi)g^E_{ab},\psi]+S^E_{\partial M}, \end{aligned}$$

where

$$K(\phi)=M_{\mathrm{Pl}}^2\left[ \frac{Z}{F}+\frac32\left(\frac{F_{,\phi}}{F}\right)^2 \right], \qquad V_E=\frac{M_{\mathrm{Pl}}^4U}{F^2}, \qquad A=\frac{M_{\mathrm{Pl}}}{\sqrt F}.$$

Where $K\gt0$, a canonical scalar $\chi$ is defined locally by $d\chi/d\phi=\sqrt K$. The gravitational sector now has the Einstein–Hilbert form, hence the name Einstein frame, but the metric followed by matter is $g^J_{ab}=A^2g^E_{ab}$. Dropping this transformed matter coupling would not be a frame change; it would replace the original theory.

Rods, clocks, and dimensionless observables

The point-particle action makes the physical content transparent:

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

Thus the Einstein-frame mass parameter carried by this action is $m_E(\chi)=A(\chi)m_J$. Varying the worldline gives

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

Massive matter is not geodesic with respect to $g^E_{ab}$ because its local mass unit varies. Equivalently, diffeomorphism invariance of the transformed matter action gives

$$\nabla^E_aT_E^{ab} =\alpha(\chi)T_E\nabla_E^b\chi, \qquad \alpha=\frac{d\ln A}{d\chi}.$$

Here $\chi$ has mass dimension one, so this $\alpha$ has mass dimension minus one. The dimensionless convention used in the scalar–tensor chapter is $\varphi=\chi/(\sqrt2 M_{\mathrm{Pl}})$ and $\alpha_{\mathrm{dimless}}=d\ln A/d\varphi =\sqrt2 M_{\mathrm{Pl}}\alpha$. Couplings must be converted together with the scalar kinetic normalization before comparing formulas between pages.

This equation does not signal a violation of total energy–momentum conservation: the scalar stress tensor carries the opposite exchange. It also does not mean that the two frames predict different trajectories. The Jordan description attributes the effect to its metric connection; the Einstein description attributes it to the scalar dependence of matter units.

If an atomic transition, a ruler, and the mass being measured all acquire the same factor, then

$$\frac{m_E}{m_{E,\mathrm{unit}}} =\frac{A m_J}{A m_{J,\mathrm{unit}}} =\frac{m_J}{m_{J,\mathrm{unit}}}.$$

The equality illustrates the right equivalence test: express the prediction as a dimensionless ratio of an observable to its realized unit and transform both. Treating the Einstein-frame mass as variable while holding the Einstein-frame laboratory unit fixed mixes two different theories. Running units, massive-particle motion, and the operational equivalence are analyzed in Faraoni and Nadeau 2007, §§3.1–3.2, pp. 7–12.

Beyond conformal transformations

A scalar gradient permits a disformal map. For functions $C(\phi)$ and $D(\phi)$,

$$\bar g_{ab}=C(\phi)g_{ab} +D(\phi)\nabla_a\phi\nabla_b\phi, \qquad X=-\frac12g^{ab}\nabla_a\phi\nabla_b\phi.$$

Algebraic inversion gives

$$\bar g^{ab} =\frac1C\left[ g^{ab}-\frac{D}{C-2DX} \nabla^a\phi\nabla^b\phi \right],$$

and in four dimensions

$$\sqrt{-\bar g} =C^2\sqrt{1-\frac{2DX}{C}}\sqrt{-g}.$$

For this map, $C\ne0$ and $C-2DX\ne0$ are necessary for a nondegenerate inverse; their signs must also preserve the desired Lorentzian signature. If $C$ or $D$ depends on $X$, nondegeneracy of the metric is not sufficient: the functional map from $(g,\phi)$ to $(\bar g,\phi)$ must itself have a nonsingular Jacobian. A singular disformal map can change the number of constraints and degrees of freedom, so it cannot be invoked as an equivalence transformation. The inverse and signature conditions, written there with the opposite sign for $X$, are given in Bettoni and Liberati 2013, §II.B, pp. 3–5, equations (6)–(14).

The terminology is best treated operationally.

Description	Gravitational kinetic sector	Metric in $S_m$	Physical statement
Jordan variables	Generally nonminimal, such as $F(\phi)R$	$g^J_{ab}$	Matter is minimally coupled by definition in the selected Jordan description.
Einstein variables	Einstein–Hilbert plus transformed extra fields	$A^2(\phi)g^E_{ab}$	Classically equivalent on an invertible domain when sources, units, and data are transformed.
New minimally coupled model	Einstein–Hilbert plus extra fields	$g^E_{ab}$	Usually a different theory from the preceding Jordan model.
Disformal variables	Depends on $C$, $D$, and operator basis	$\bar g_{ab}$ or its transformed expression	Equivalence requires a nonsingular map and the full transformed matter and boundary sectors.

Bettoni and Liberati compare Jordan, Einstein, Galileon, and disformal frames with the matter metric kept explicit in Bettoni and Liberati 2013, §IV, pp. 7–9, equations (33)–(39).

Where equivalence can fail

Classical frame equivalence has a precise, limited domain.

• The map must be smooth, one-to-one, and invertible on the entire solution branch being compared. A surface with $F=0$, $K=0$, or $C-2DX=0$ is not crossed by the preceding argument.
• Initial, boundary, asymptotic, and regularity data must map bijectively. A transformation that sends finite data to singular data can remove a solution even when it is invertible in a smaller local patch.
• Matter fields, external sources, composite operators, units, and detectors must transform. An off-shell Green function need not be invariant even when the associated on-shell observable is.
• In a quantum treatment the functional measure, regulator, renormalization prescription, and possible anomaly must be transformed consistently. Classical equivalence alone is not a proof of equality for truncated quantum effective actions.

The first three points can be checked inside the classical theory. The last is a separate quantum claim and must not be inferred from a classical change of variables.

Common pitfalls

“Einstein frame” identifies the physical metric. The label identifies a canonical gravitational kinetic term. The physical measuring metric is the one in the matter action and may be $A^2g_E$, not $g_E$.

Dimensionful numbers must agree between frames. A mass, time, or length can scale under a field redefinition. A consistently realized dimensionless ratio is the appropriate invariant comparison.

A nonsingular metric determinant proves equivalence. For derivative- or $X$-dependent transformations, the full field-space Jacobian and the mapping of data must be nonsingular as well.

Exercises

Starting from $S_{\mathrm{pp}}=-m_J\int ds_J$ and $g^J_{ab}=A^2(\chi)g^E_{ab}$, derive the Einstein-frame equation of motion. Then show why the dimensionless ratio of the particle mass to an atomic mass standard is frame independent when both arise from the same universal matter coupling.

Solution

Because $ds_J=A\,ds_E$, the action is $-\int m_E(\chi)ds_E$ with $m_E=A m_J$. Vary a parametrized worldline and then choose Einstein-frame proper time. The tangential part fixes the evolution of $m_E$; projecting orthogonally to $u_E^a$ gives

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

Dividing by $m_E$ yields

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

since $m_J$ is constant in the Jordan description. The right-hand side is orthogonal to $u_E^a$, as required by $u_E^au^E_a=-1$.

Let an atomic reference mass be $m_{J,0}$. Universal coupling gives $m_{E,0}=A m_{J,0}$ at the same event. Therefore

$$\frac{m_E}{m_{E,0}} =\frac{A m_J}{A m_{J,0}} =\frac{m_J}{m_{J,0}}.$$

The same cancellation occurs in frequency ratios and proper-time comparisons once the clock realization is transformed. If instead one declares $m_{E,0}$ constant while retaining $m_E=A m_J$, one has changed the matter coupling and is comparing a new model, not two frames of the original one.

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.
• Faraoni, Valerio, and Shahn Nadeau (2007), “The (Pseudo)issue of the Conformal Frame Revisited,” Physical Review D 75, 023501. DOI; Open PDF.