Theory Space and Effective Descriptions

Before asking whether an alternative theory predicts a new black hole or a different cosmology, one must know what the theory is and how its approximation is organized. This chapter supplies that common language. It separates a change of physical theory from a change of field variables, orders operators by invariant expansion parameters, removes redundant descriptions, defines a controlled GR limit, and assembles the result into one comparison record that the rest of the volume will reuse.

Helpful background. Einstein Equations provides the baseline equations and Bianchi identity; Gravitational Actions and Boundary Terms provides the complete action principle needed to compare descriptions; Formulations of General Relativity provides the constraint, gauge, and initial-value language used to compare solution spaces.

Enter this chapter

The chapter’s central object is the complete theory specification

$$\mathfrak T=\bigl(M,\Phi,\mathcal G,S,\mathcal B,\mathcal O\bigr),$$

where the entries record spacetime and global structure, dynamical fields, gauge and global symmetries, the boundary-completed action, admissible data and branches, and physical observables including the matter metric. The tuple is not an exercise in notation. It prevents four common category errors:

• treating equal-looking bulk equations as equal boundary-value problems;
• treating a conformal or disformal field map as physical equivalence after leaving the matter sector fixed;
• treating every symmetry-allowed operator as equally important without a cutoff and power counting; and
• treating a parameter that tends to zero as proof that sourced solutions and observables approach GR.

The six pages solve these problems in an order that mirrors an actual theory calculation.

Page	Question answered	Result carried forward
Actions, Fields, and Symmetries	Which data distinguish a theory from a reformulation?	A boundary-complete variational problem, gauge identity, matter metric, admissible solution space, and observable map.
Jordan, Einstein, and Other Field Frames	When does a field change preserve observables?	Smooth invertibility plus transformed matter, sources, units, and data; singular maps and changed matter couplings are excluded from the equivalence claim.
Scales, Operators, and Power Counting	Which terms matter at a target accuracy?	Canonically normalized operator, derivative, curvature, loop, threshold, and secular expansion parameters with a first-omitted-order error.
Field Redefinitions and Redundant Operators	Why can different actions give the same on-shell result?	A nonredundant basis obtained with local perturbative maps while retaining coefficient, source, boundary, Jacobian, and higher-order shifts.
General-Relativistic Limits and Decoupling Limits	What must converge for GR to be recovered?	A scaling path on a named branch where modes, data, sources, dimensionless observables, and validity domain converge together.
Comparing Gravitational Theories	Which common questions must every theory answer?	An eleven-field comparison that distinguishes known results, explicit failures, and sharply stated open calculations.

The order matters. Power counting before canonical normalization can assign the wrong suppression scale. Power counting before basis reduction can promote a redundant operator. Taking a GR limit before fixing the source map can hide a finite helicity-0 coupling. Comparing phenomenology before these steps can mistake a solution in an inconsistent or uncontrolled theory for a prediction.

One calculation, six views

Metric $f(R)$ gravity illustrates how the pages fit together. Begin with

$$S=\frac{1}{16\pi G}\int d^4x\sqrt{-g}\,f(R)+S_m[g,\psi] +S_{\partial M}.$$

The theory record says that $g_{ab}$ is varied, matter sees $g_{ab}$, and the boundary term and allowed data must be stated. On a domain where $f_{RR}\ne0$, the Legendre variable $\varphi=f_R$ and a conformal metric can rewrite the model as Einstein gravity plus a scalar with transformed matter coupling. The frame page explains why this is one classical theory only where the map is invertible and the matter metric is carried along.

The power-counting page then asks whether $f(R)=R+\alpha R^2+\cdots$ is a truncated curvature EFT or a fundamental scalar–tensor action. Those are different claims at frequencies comparable to the extra scalar mass. The field-redefinition page checks which curvature combinations are independent in the stated dimension and matter sector. The trace equation

$$3\Box f_R+f_RR-2f=8\pi G T$$

exposes the metric scalar degree of freedom. Finally, the GR-limit page asks whether the scalar mass moves above all used scales and whether its boundary data remain on the GR-connected branch as $\alpha\to0$. The comparison record keeps this metric theory distinct from Palatini $f(\mathcal R)$, where the analogous trace equation is algebraic. The standard metric and Palatini derivations are collected in Sotiriou and Faraoni 2010, §§II–III, pp. 5–15.

Conventions that carry forward

All pages use metric signature $(-,+,+,+)$, the site curvature convention, $c=\hbar=k_{\mathrm B}=1$, explicit $G$, and $M_{\mathrm{Pl}}^{-2}=8\pi G$ when the reduced Planck mass is useful. A source with another sign for the scalar kinetic variable, Riemann tensor, or conformal map is translated as a complete package. Boundary and total-divergence terms remain visible until the variational data are fixed.

Four declarations recur in later chapters:

1. Matter metric: which metric or frame defines stress tensors, geodesics, rods, clocks, masses, and dimensionless observables?
2. Branch and data: which algebraic root, asymptotic sector, center or horizon condition, and Green-function prescription selects the solution?
3. Canonical modes: after all constraints, which variables propagate and with what kinetic normalization on the named background?
4. Validity: which local energies, curvatures, gradients, amplitudes, secular times, and loop factors are small relative to which cutoff?

If one of these declarations is absent, a later stability or phenomenology claim cannot be transported safely.

Choose a route

Construct a new EFT. Read the action page, power counting, field redefinitions, and the comparative record. List the full symmetry-allowed operator basis before calculating a favorite correction, then bound the first omitted order.

Translate an existing model. Read the action page and frame page, then record the matter metric, units, boundary data, and singular surfaces in both descriptions. Use one dimensionless observable as an independent check.

Test a claimed GR limit. Read power counting and field redefinitions before the limits page. Count modes, hold canonical sources fixed, state the order of limits, and demand uniform convergence on a domain below the cutoff.

Prepare for the consistency chapter. Complete the comparison record through equations, constraints, modes, and cutoff. The next chapter supplies the Hamiltonian, kinetic, dispersion, hyperbolicity, strong-coupling, and radiative-stability tests needed to fill those entries honestly.

Review the chapter

Two authors present four-dimensional actions related by

$$g'_{ab}=\Omega^2(\phi)g_{ab} +\frac{c}{\Lambda^2}R_{ab},$$

and claim that their theories are physically identical. Give a sequence of checks that could establish the claim through $O(\Lambda^{-2})$, and identify three ways it could fail.

Solution

First write the complete theory record on both sides: dynamical fields, nondynamical structures, gauge symmetries, bulk and boundary action, initial and asymptotic data, matter metric, branch, cutoff, and observables. Verify that $\Omega$ is finite and nonzero and that the full field-space map, including the derivative metric term, is perturbatively invertible throughout the branch. Transform the gravitational boundary completion as well as the bulk density.

Next transform the matter action, external sources, composite operators, units, and the selected observable. Expand the action consistently through $O(\Lambda^{-2})$. The $R_{ab}$ term moves lower-order Einstein equations into higher operators, so record all induced curvature, stress-tensor, and boundary couplings. Reduce both descriptions to nonredundant bases at the same order and match their Wilson coefficients. Check the functional Jacobian in a quantum calculation. Canonically normalize the physical modes on the same background, map the initial and boundary data, and verify an on-shell dimensionless observable independently in both variables. The discrepancy should be no larger than the first omitted $O(\Lambda^{-4})$ terms.

The claim fails if the conformal factor or full derivative map becomes singular, if one author keeps matter minimally coupled to $g'$ instead of transforming the original matter metric, or if the transformation changes boundary or asymptotic data. It can also fail if one calculation truncates induced terms inconsistently, crosses a different solution branch, compares an off-shell Green function as if it were invariant, or ignores a nontrivial quantum Jacobian or anomaly. Agreement of the written vacuum bulk equations is therefore necessary in some comparisons but never sufficient.

References

• Burgess, C. P. (2004), “Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory,” Living Reviews in Relativity 7, article 5. DOI; Open PDF.
• Criado, Juan Carlos, and Manuel Pérez-Victoria (2019), “Field Redefinitions in Effective Theories at Higher Orders,” JHEP 03, 038. DOI; Open PDF.
• Iyer, Vivek, and Robert M. Wald (1994), “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy,” Physical Review D 50, 846–864. DOI; Open PDF.
• Sotiriou, Thomas P., and Valerio Faraoni (2010), “f(R) Theories of Gravity,” Reviews of Modern Physics 82, 451–497. DOI; Open PDF.