Consistency Tests and Failure Diagnosis

The sharpest consistency test is the one whose failure has a distinctive scaling while all other inputs are held fixed. No single limit wins universally. A Newtonian or test-mass check can falsify an analytic coefficient; resolution scaling can isolate numerical error; a first law can expose an inconsistent conservative dictionary; and flux balance can localize a dissipative sign or normalization. The general method is to translate every framework to a shared invariant, vary one regime parameter at a time, and identify the first failed assumption.

Helpful background. The Einstein Equation supplies exact identities; Weak Metric Perturbations and Gauge Freedom supplies Newtonian and gauge checks; Presymplectic Potentials and Currents supplies canonical equivalence; Computational Relativity supplies convergence tests; Gravitational Radiation and Waveforms supplies asymptotic balance observables; and Relativistic Two-Body Dynamics supplies the approximation domains compared below.

A hierarchy of invariant checks

Begin with the cheapest tests. A high-level comparison should not be attempted while a lower-level identity fails.

1. Definitions and dimensions. Fix $M$, $\mu$, $\nu$, spin normalization, $E$, $J$, frequency, time variable, and powers of $G$. A dimensionful logarithm or unequal normalization invalidates later comparisons.
2. Algebraic symmetries. Test exchange $m_1\leftrightarrow m_2$, parity, time reversal in the conservative sector, and the spinless or zero-tide limit.
3. Exact limiting dynamics. Recover Newtonian scattering or binding, Schwarzschild/Kerr geodesic motion as $\nu\to0$, and zero interaction as $G\to0$.
4. Variational and conservation identities. Check the binary first law for conservative circular data and energy–angular-momentum balance for dissipative data.
5. Common asymptotic coefficients. Double-expand PN and PM results, or expand self-force and PN results in their overlap.
6. Independent numerical evidence. Refine resolution, extraction radius, basis, regularization, and calibration split before interpreting a residual as missing physics.

Every method must enter through a common observable:

Framework	Native intermediate object	Preferred shared invariant
PN	Coordinate acceleration or canonical Hamiltonian	$E_b(\Omega)$, periastron advance, or $\chi(E,J)$
PM and amplitudes	$G$-series, impulse kernel, or on-shell amplitude	$\chi(E,J)$, impulse, and radiated four-momentum
Self-force	Regularized field or force component	Redshift, frequency ratios, spin precession, tidal eigenvalues, or $\chi(E,J)$
EOB	Potentials and resummed Hamiltonian	The same binding, scattering, and frequency invariants
Numerical relativity	Coordinates, extracted modes, and horizon data	Asymptotic fluxes, remnant charges, binding curves, or scattering angle with error bars

This translation prevents a canonical or gauge difference from masquerading as a physical failure. Damour’s scattering dictionary, for example, compares the real and effective dynamics through the function $\chi(E,J)$ rather than through coordinate potentials Damour 2016, §§II–III and V–VIII, PDF. The radial action provides the corresponding Hamilton–Jacobi bridge for scattering and bound observables Kälin and Porto 2020, §§2 and 5, pp. 6–8 and 17–30, PDF.

Limiting-case fingerprints

Different limits probe different pieces of a calculation.

Newtonian limit. At $v^2\ll1$ and $GM/r\ll1$, binding energy, Kepler’s law, Rutherford-type deflection, and quadrupolar balance fix signs and leading normalizations. Failure here usually indicates masses, $G$, time, or state normalization rather than strong-field physics.

Test-mass limit. At fixed invariant orbital variables and $\nu\to0$, nonspinning EOB dynamics must reduce to geodesic motion in the appropriate background. A residual that tends to a nonzero constant as $\nu\to0$ points to the background, energy map, or variable conversion. A residual proportional to $\nu^2$ instead suggests a missing second-order self-force term after the linear coefficient has been matched.

High-energy and weak-deflection limit. PM and amplitude predictions should agree at fixed center-of-mass energy and large $J$. This tests relativistic normalization and iteration subtraction. It does not validate a conservative approximation once radiated energy is comparable to the desired accuracy.

Circular and eccentric limits. A generic-orbit Hamiltonian should reduce smoothly to circular binding and small-eccentricity periastron advance away from a separatrix. Terms multiplying radial momentum must vanish appropriately, while derivatives of those terms can remain visible in radial frequencies.

Spin and finite-size limits. Spin-orbit terms are odd in a single spin and spin-spin terms are even under simultaneous reversal. Tidal terms must vanish with their response coefficients, not merely with coordinate deformation. Exchange symmetry should be tested using $\nu$ and symmetric/antisymmetric spin combinations.

First-law and balance triangulation

For conservative circular, nonspinning point-mass binaries, with body structure held fixed and $E$ including both rest masses, the first law has the form

$$\delta E-\Omega\,\delta J =z_1\,\delta m_1+z_2\,\delta m_2,$$

with the redshifts and asymptotic time normalization defined consistently. For binding energy $E_b=E-m_1-m_2$, replace each $z_A$ on the right by $z_A-1$. The law triangulates three computations: a Hamiltonian gives $E$ and $J$, self-force gives a redshift coefficient, and EOB supplies a finite-ratio representation. Le Tiec, Blanchet, and Whiting derive the relation in PN theory and general relativity and use it to link gauge-invariant circular quantities Le Tiec, Blanchet, and Whiting 2012, §§II–IV, especially Eqs. (2.13), (3.8), and (4.1). If $E(\Omega)$ and $J(\Omega)$ agree but the law fails, inspect the mass held fixed in the derivative and the conversion between $q$ and $\nu$ before altering the dynamics.

For dissipative evolution,

$$\frac{dE_{\rm mech}}{dt}=-\mathcal F_E, \qquad \frac{dJ_{\rm mech}}{dt}=-\mathcal F_J,$$

up to declared Schott or near-zone storage terms. On an exactly circular adiabatic sequence, $\mathcal F_E=\Omega\mathcal F_J$. A local reaction force, an asymptotic flux, and the secular change of orbital invariants then form an independent triangle. Kosower, Maybee, and O’Connell formulate impulse and radiated momentum from the scattering matrix and check total momentum conservation explicitly Kosower, Maybee, and O’Connell 2019, §§3.2–3.5 and 5.3–5.4, pp. 8–16 and 46–49, PDF. A balance failure that reverses sign under retarded versus advanced boundary conditions identifies a dissipative prescription, not a conservative Hamiltonian coefficient.

Hereditary tails require care because the mechanical energy may contain nonlocal contributions. Blanchet’s matched multipolar construction separates near-zone, radiation-zone, instantaneous, and hereditary pieces Blanchet 2024, §2, especially §§2.2–2.4. Testing only a local force against a total flux can leave a scale-dependent remainder that cancels in the complete balance law.

Residual scaling as a diagnostic

Model a residual in one invariant as

$$\Delta Q(x,\nu,h,R) =a\,x^{N+1} +b\,\nu^{k+1} +c\,h^p +d\,R^{-s} +e\,\Delta_{\rm cal} +\cdots,$$

where $x$ is a weak-field parameter, $h$ a numerical resolution scale, $R$ an extraction radius, and $\Delta_{\rm cal}$ a calibrated-basis discrepancy. This equation is a diagnostic ansatz, not a claim that all errors are independent.

Vary $h$ at fixed physical inputs to estimate $p$; vary $R$ only after the bulk solution is resolved; vary $x$ within the common asymptotic region; vary $\nu$ while preserving the same invariant definition. A residual proportional to $h^p$ is numerical. A continuum residual proportional to the first omitted $x$ power is analytic truncation. A residual proportional to $\nu^{k+1}$ after matching through $\nu^k$ points to missing mass-ratio order. A residual that changes under a canonical transformation while $\chi$ does not is representational.

The numerical series must actually converge. The SXS catalog labels a configuration nonconvergent when the highest-versus-next resolution mismatch does not improve over the preceding pair and notes that different resolutions can have slightly different physical parameters Boyle et al. 2019, §4, pp. 24–28, PDF. Fitting a formal power of $h$ to such points produces a precise but meaningless continuum estimate.

Failure localization matrix

Observed symptom	Most discriminating next test	Likely source if the test fails
Constant factor at all regimes	Dimensions and tree/Newtonian limit	State, mass, Fourier, or flux normalization
Difference disappears in $\chi$ but not in $H$	Canonical transformation of actions	Representation, not physics
Agreement at $\nu=0$ but residual scales as $\nu^2$	First- and second-order self-force expansion	Missing finite-mass-ratio order
PN–PM mismatch only at low velocity	Hold $GM/(bv^2)$ fixed and small	Nonuniform limit or wrong double expansion
Local reaction disagrees with flux	Include Schott and hereditary terms	Boundary condition or incomplete balance
Training agreement, blind failure	Grouped held-out calibration test	Overfitting or extrapolation
Difference shrinks as $h^p$	Independent resolution series	Numerical discretization
Difference persists across independent representations	Exact limit and third-method triangulation	Candidate physical discrepancy

This matrix orders tests; it does not replace judgment. A third method is informative only if it does not inherit the same input or data. Three EOB implementations calibrated to one catalog are not three independent checks of that catalog.

Noncommuting limits

The order of limits is itself part of the diagnosis. In PN–PM matching, $v\to0$ at fixed $G$ and $b$ leaves the weak-deflection domain because $GM/(bv^2)$ grows; the PM truncation is then nonuniform. In self-force theory, approaching a light ring at fixed finite $\nu$ invalidates an expansion justified by taking $\nu\to0$ at fixed orbit. Near a separatrix, the small-eccentricity and long-time limits can amplify radial-period logarithms. In numerics, $h\to0$ at fixed finite extraction radius does not remove finite-radius bias, and $R\to\infty$ at unresolved $h$ does not establish a continuum result.

Therefore record a path through parameter space, not just endpoint labels such as “Newtonian,” “test mass,” or “continuum.” Apparent contradictions often disappear when both calculations are evaluated on the same path.

Solved two-stage failure diagnosis

At fixed $x$ and $\nu=0.10$, an illustrative EOB–numerical residual in a dimensionless binding invariant is measured at three grid scales. These are constructed exercise data, not results from a numerical-relativity run:

$h/h_0$	Residual
1	$5.0\times10^{-3}$
$1/2$	$1.4\times10^{-3}$
$1/4$	$5.0\times10^{-4}$

After continuum extrapolation, a nonzero residual is found at three symmetric mass ratios:

$\nu$	Continuum residual
0.10	$2.0\times10^{-4}$
0.20	$8.0\times10^{-4}$
0.25	$1.25\times10^{-3}$

Assume the model already contains the exact test-mass and first-order self-force terms. Diagnose both scalings.

Solution

Let $D(h)=D_0+Ch^p$. Differences remove the unknown continuum residual:

$$\frac{D(h)-D(h/2)}{D(h/2)-D(h/4)} =\frac{0.0050-0.0014}{0.0014-0.0005}=4=2^p.$$

Thus $p=2$. Richardson extrapolation gives $D_0=[4D(h/4)-D(h/2)]/3=2.0\times10^{-4}$, agreeing with the first row of the second table. Dividing the raw residuals would give the wrong order because they include this nonzero offset.

The continuum values satisfy

$$\frac{\Delta Q}{\nu^2}=0.020$$

for all three mass ratios. Because the $\nu^0$ and $\nu^1$ pieces are already fixed, the remaining fingerprint is compatible with a missing or incorrect $O(\nu^2)$ coefficient. It does not uniquely identify the source: a missing PN term or calibration bias proportional to $\nu^2$ would have the same mass-ratio scaling. A second-order self-force calculation and an independent sweep in $x$ would separate these hypotheses. Three constructed points illustrate the diagnostic; they do not establish a real model’s error budget.

Once the source is localized, its magnitude belongs in the cross-regime uncertainty budget without double counting the raw residual.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27, 4 (2024). DOI.
• Boyle, Michael, et al. “The SXS Collaboration Catalog of Binary Black Hole Simulations.” Classical and Quantum Gravity 36, 195006 (2019). DOI. Open PDF.
• Damour, Thibault. “Gravitational Scattering, Post-Minkowskian Approximation and Effective One-Body Theory.” Physical Review D 94, 104015 (2016). DOI. Open PDF.
• Kälin, Gregor, and Rafael A. Porto. “From Boundary Data to Bound States.” Journal of High Energy Physics 2020, 072 (2020). DOI. Open PDF.
• Kosower, David A., Ben Maybee, and Donal O’Connell. “Amplitudes, Observables, and Classical Scattering.” Journal of High Energy Physics 2019, 137 (2019). DOI. Open PDF.
• Le Tiec, Alexandre, Luc Blanchet, and Bernard F. Whiting. “The First Law of Binary Black Hole Mechanics in General Relativity and Post-Newtonian Theory.” Physical Review D 85, 064039 (2012). DOI.