Degrees of Freedom and Dynamical Consistency

A modified gravitational action becomes a candidate physical theory only after its propagating content and evolution are shown to be controlled. This chapter supplies that consistency chain. It first counts physical modes, then tests their kinetic energy and dispersion, asks whether the complete gauge-fixed equations possess a well-posed Cauchy problem, identifies the scale where interactions defeat perturbation theory, and finally checks whether quantum corrections preserve the assumed operator hierarchy. Passing one stage never substitutes for the next.

Helpful background. Einstein Equations provide the healthy two-tensor baseline, Gravitational Actions and Boundary Terms fix the variational problem, Formulations of General Relativity supply gauge and constraint decompositions, and Variational and Symplectic Geometry identifies reduced physical directions.

The dynamical-consistency chain

Every conclusion in this chapter belongs to five declared ingredients: a covariant action, a background, a gauge-fixed formulation, a solution branch, and an EFT regime. A result obtained after changing one ingredient cannot silently be transferred to the others. In particular, a symmetric background can acquire accidental constraints, a gauge choice can spoil an otherwise healthy evolution system, and a branch-dependent kinetic coefficient can move the strong-coupling scale.

The six pages form a progressive sequence:

1. Hamiltonian Constraint Counting runs the Dirac algorithm and derives $N_{\mathrm{dof}}=N_q-N_{\mathrm{FC}}-N_{\mathrm{SC}}/2$. It verifies the convention with both ADM bookkeeping and the Proca second-class pair.
2. Ghosts and Kinetic-Sign Instabilities reduces the constraints before diagnosing negative kinetic eigenvalues, wrong-sign pole residues, or an Ostrogradsky mode.
3. Gradient, Tachyonic, and Jeans Instabilities distinguishes ultraviolet-fast gradient growth from finite tachyonic growth and physical long-wavelength Jeans collapse.
4. Hyperbolicity and the Cauchy Problem states a bounded-symmetrizer theorem, follows its energy estimate, and uses a Jordan block to show why real characteristics alone are insufficient.
5. Strong Coupling and Predictive Cutoffs canonically normalizes interactions and uses the cubic Galileon to separate a nonlinear screened background from the fluctuation cutoff.
6. Matter Coupling and Radiative Stability compares symmetry protection, loop-generated operators, and unresolved theory dependence at the evidence cutoff 2026-08-19.

This order is logical rather than cosmetic. There is no physical kinetic matrix until gauge and constraints have been removed; no trustworthy dispersion relation until the kinetic matrix is invertible; no nonlinear prediction until the Cauchy problem and cutoff are controlled; and no durable tuned hierarchy until its radiative corrections have been estimated.

One quadratic action, several distinct tests

After reduction, the middle three pages share the local action

$$S_{\mathrm{red}}^{(2)}=\frac12\int dt\,d^3x\,a^3 \left[ \dot q^{\mathsf T}K\dot q -\frac{1}{a^2}\partial_iq^{\mathsf T}G\partial_iq -q^{\mathsf T}M^2q +2\dot q^{\mathsf T}Bq \right].$$

The matrix $K$ tests kinetic inertia. Once $K$ is positive and canonical, generalized eigenvalues of $Gv=c_s^2Kv$ test high-frequency gradients, while the mass matrix and background duration determine tachyonic growth. These are properties of physical perturbations. Strong hyperbolicity is broader: it tests the full principal symbol, including gauge drivers and constraint propagation, and requires a uniformly bounded characteristic basis.

The distinction is visible in the sources. The canonical constraint structure of general relativity is developed in Arnowitt, Deser, and Misner 1962/2008, §§ 3.4 and 4.2–4.6. Papallo and Reall’s generalized-harmonic obstruction and the later modified-harmonic construction demonstrate that formulation choice matters; the constructive weak-coupling result is given in Kovacs and Reall 2020, Appendix A and §§ 4.1–4.2.

What each successful test does not prove

Successful result	What it establishes	What remains open
Correct constraint count	Dimension of the local reduced phase space	Kinetic sign, characteristics, nonlinear scale, and branch regularity
Positive reduced kinetic matrix	No quadratic ghost in the tested sector and regime	Positive gradients, full-system hyperbolicity, or radiative protection
Positive high-frequency sound speeds	No local gradient growth in the resolved physical modes	Slow tachyons, Jeans growth, gauge blocks, or causal completion
Bounded full-symbol symmetrizer	A strongly hyperbolic formulation under its regularity and boundary hypotheses	Global existence, another branch, or quantum stability
Suppressed canonical interactions	Perturbative fluctuation control at the stated background scale	Suppression of every omitted EFT operator or heavy-state threshold
Technically natural coefficients	Loop corrections respect the declared hierarchy in the computed sectors	A UV completion or protection under arbitrary matter coupling

The table prevents a common category error: “ghost free” is sometimes used as shorthand for all consistency. It is only one gate. A theory can have the intended number of modes and positive energy yet fail because its evolution is weakly hyperbolic. It can also be locally well posed but predictive only over a very narrow scale interval.

A worked synthesis

Consider a scalar extension on a homogeneous branch. Suppose the completed constraint algebra gives the two tensor polarizations plus one scalar. After integrating out lapse and shift, the scalar kinetic coefficient is $Z\gt0$, its sound speed satisfies $c_s^2=0.3$, and its effective mass is $m_{\mathrm{eff}}^2=-0.01H^2$. These facts establish no ghost and no high-frequency gradient instability. They do not require rejecting the branch: the tachyonic rate is slower than the background expansion scale and must be compared with the duration and initial-data class.

Next suppose a modified-harmonic first-order reduction has a uniformly bounded symmetrizer while higher-derivative corrections remain perturbative. That supplies a local well-posedness result for that formulation, not for arbitrary gauges or a strong-field branch. A large background kinetic factor $Z=100$ raises a cubic fluctuation scale schematically by $Z^{1/2}=10$, but omitted operators and background gradients can still impose a lower cutoff. Finally, loop power counting must show that corrections to the degeneracy and matter coupling remain below the accuracy target. If a small correction changes the constraint rank, coefficient smallness is not enough.

Independent limits check the whole synthesis. In the controlled GR limit the extra scalar must decouple or remain a healthy separate mode, the two Einstein tensor modes retain their standard positive kinetic term, and the gauge-fixed principal symbol approaches a known strongly hyperbolic GR formulation. In the vacuum-background limit, any redressed scale returns to its unredressed value. These checks expose mismatched normalizations and singular branch limits.

Boundaries of the chapter

The chapter evaluates actions, modes, formulations, branches, cutoffs, and radiative stability. It does not perform empirical inference. Instrument systematics, likelihoods, and mutable numerical bounds belong to Tests of Gravity. Stable low-energy quantum general relativity belongs to Quantum Gravity, while generic quantum-field renormalization and PDE theory receive only the bounded bridges required here.

A theory that passes this chapter has earned the right to make controlled predictions in a stated regime; it has not earned a universal verdict. Later chapters repeat these tests for scalar–tensor, higher-curvature, vector–tensor, affine, massive, extra-dimensional, nonlocal, and low-acceleration models, because the matrices, constraints, and branches are theory dependent.

Common pitfalls

Skipping directly from an action to observables. Field content, constraint reduction, kinetic signs, and the Cauchy problem must be fixed before a waveform or cosmological parameter is interpreted.

Treating a symmetric background as generic. Accidental degeneracy can hide a mode whose kinetic term returns on nearby backgrounds, often at a low strong-coupling scale.

Using “stable” without a scale. State the wavenumber interval, growth time, branch, background duration, and cutoff that make the verdict meaningful.

Review the chapter

A proposed branch has $N_q=5$, two first-class constraints, and two second-class constraints. Its reduced scalar sector has $K=2$, $G=-0.08$, and a cutoff $\Lambda$. The authors report real characteristic speeds for the unreduced gauge-fixed equations but do not exhibit a complete eigenbasis. Classify what is established and what fails.

Solution

The constraint count gives $N_{\mathrm{dof}}=5-2-1=2$. The positive scalar kinetic coefficient excludes a quadratic ghost in that reduced sector. However, $c_s^2=G/K=-0.04$, so subcutoff high-wavenumber modes have gradient growth rate $0.2k_{\mathrm{phys}}$ and the branch fails wherever this sign persists inside the EFT window. There is also an inconsistency in the reported checks: a faithful full evolution system must retain this physical scalar characteristic, whose speed is imaginary. It cannot have an entirely real characteristic spectrum on the same background and in the same regime. The reduction, the chosen data, or the full-symbol calculation must therefore be rechecked. Even after that contradiction is resolved, real characteristic speeds alone would not establish strong hyperbolicity without a uniformly bounded complete eigenbasis, as the Jordan-block example shows. No claim about radiative stability follows until loop-generated operators and matter couplings are checked.

References

• Arnowitt, Richard, Stanley Deser, and Charles W. Misner. 1962/2008. “The Dynamics of General Relativity.” In Gravitation: An Introduction to Current Research, edited by Louis Witten, 227–265. arXiv:gr-qc/0405109.
• Date, Ghanashyam. 2010. “Lectures on Constrained Systems.” arXiv:1010.2062.
• Kovacs, Aron D., and Harvey S. Reall. 2020. “Well-Posed Formulation of Lovelock and Horndeski Theories.” Physical Review D 101, 124003. DOI; arXiv:2003.08398.
• Woodard, Richard P. 2015. “The Theorem of Ostrogradsky.” Scholarpedia 10, 32243. DOI; arXiv:1506.02210.