Gauge-Invariant Energetics and Frequencies

Coordinates, gauges, field variables, and canonical phase-space variables are indispensable calculational tools, but their numerical values need not be physical observables. Two compact-binary calculations can look different and still describe the same dynamics. The clean comparison is made only after both are expressed in the same invariant input data and the same gauge-invariant output.

The validity regime throughout this page is asymptotically flat general relativity with transformations that preserve the stated asymptotic frame, regularity, and boundary conditions. More general gauge transformations can change quantities that are invariant only within a helically symmetric or perturbative class.

Required background. The Einstein Equation supplies the covariant dynamics, and Weak Metric Perturbations and Gauge Freedom distinguishes a metric perturbation from its curvature and observable consequences.

Helpful background. Presymplectic Potentials and Currents provides the origin of charges and Hamiltonian generators, while Black-Hole Perturbation Theory develops gauge-invariant quantities on a fixed black-hole background.

Circular-orbit energy and frequency

Consider a conservative, nonprecessing binary with a helical symmetry approximation. The coordinate separation $r$, coordinate velocity, and coefficients of a Hamiltonian depend on gauge or canonical variables. The orbital frequency $\Omega$, normalized with respect to asymptotic time, and the conserved energy $E$ associated with the same asymptotic time translation are suitable comparison quantities.

Define

$$x=(GM\Omega)^{2/3}, \qquad e(x)=\frac{E(x)-M}{\mu}.$$

At Newtonian order,

$$E_{\mathrm N}-M=-\frac{Gm_1m_2}{2r}, \qquad \Omega^2=\frac{GM}{r^3},$$

so elimination of $r$ gives

$$e_{\mathrm N}(x)=-\frac{x}{2}.$$

This elimination is the essential move. Either input relation can change under a redefinition of $r$, but the composite relation $E(\Omega)$ does not. At first post-Newtonian order in general relativity,

$$e(x)=-\frac{x}{2} \left[ 1-\left(\frac34+\frac{\nu}{12}\right)x+O(x^2) \right].$$

The test-mass check fixes both sign and normalization. For a particle on a circular Schwarzschild orbit, with $u=GM/r$,

$$\frac{E_{\mathrm{geo}}}{\mu} =\frac{1-2u}{\sqrt{1-3u}}, \qquad x=u.$$

Expanding and subtracting the rest mass gives

$$\frac{E_{\mathrm{geo}}-\mu}{\mu} =-\frac{x}{2}+\frac{3x^2}{8}+O(x^3),$$

which agrees with the $\nu\to0$ limit above. Circular PN energetics, including its gauge-invariant formulation, is reviewed in Blanchet 2024, §3.2.5, Eqs. (375)–(378).

An energy–frequency relation presumes conservative motion. In a real inspiral, $\Omega$ changes slowly and $E(\Omega)$ is used instantaneously within an adiabatic or post-adiabatic construction. It should not be mistaken for a globally conserved energy along the dissipative trajectory.

Fundamental frequencies and periastron advance

An eccentric bound orbit has distinct radial and azimuthal periods. With asymptotically normalized coordinate time,

$$\Omega_r=\frac{2\pi}{T_r}, \qquad \Omega_\phi=\frac{\Delta\phi}{T_r}, \qquad K=\frac{\Omega_\phi}{\Omega_r}.$$

The periastron advance per radial period is

$$\Delta\Phi=2\pi(K-1).$$

Although a coordinate representation of the orbit changes under a regular gauge transformation, the accumulated angle between successive radial turning points and the corresponding frequency ratio are invariant when the transformation preserves the asymptotic frame and periodic identification. In the weak-field, small-eccentricity limit,

$$K=1+\frac{3GM}{p}+O\!\left(\frac{G^2M^2}{p^2}\right),$$

where $p$ can be replaced by invariant energy and angular momentum before making a cross-gauge comparison.

Generic bound Kerr motion has radial, polar, and azimuthal frequencies. Frequency ratios expose resonances and supply inputs to action-angle perturbation theory. The frequencies must all refer to the same time parameter: proper-time, Mino-time, and asymptotic-time frequencies are different quantities connected by explicit conversion factors.

Redshift and precession invariants

At small mass ratio, a particularly useful circular-orbit observable is the redshift variable

$$z=\frac{d\tau}{dt}=\frac{1}{u^t}$$

evaluated on the regularized worldline in a helically symmetric gauge. It has a first-order expansion

$$z(x,\nu)=z_0(x)+\nu z_1(x)+O(\nu^2).$$

The coefficient $z_1$ can be computed by self-force methods and related, through a first law of binary mechanics, to derivatives of the binding energy and angular momentum. It is not invariant under arbitrary gauge transformations: the permitted class must respect helical symmetry, regularity, and the asymptotic normalization of time. This qualified notion of invariance is explained in Barack and Pound 2019, §6.2.

Spin-precession invariants compare the accumulated rotation of a transported spin with the orbital phase. Tidal eigenvalues compare the electric and magnetic parts of curvature in a specified comoving frame. These quantities carry more local strong-field information than $E(\Omega)$, but their definitions include a worldline, tetrad, and regularization prescription. “Gauge invariant” never means “definition free.”

Scattering observables

For an unbound encounter, asymptotic energy $E$ and angular momentum $J$ replace orbital frequency as inputs. The conservative scattering angle is

$$\chi(E,J)=\Delta\phi(E,J)-\pi.$$

For local, central conservative dynamics define the full, unnormalized radial action with a large-radius cutoff,

$$I_r(E,J;R)=2\int_{r_{\min}}^R p_r(E,J;r)\,dr.$$

Then

$$\chi(E,J)=-\lim_{R\to\infty}\frac{\partial I_r(E,J;R)}{\partial J}-\pi.$$

The derivative is taken before removing the cutoff: the action itself has large-distance divergences, although the angle is finite. The $-\pi$ removes the straight-line azimuthal change; it is omitted if one instead differentiates the interaction action after subtracting the free action. A further energy-dependent subtraction leaves the angle unchanged. See the radial-action derivation for endpoint and infrared details. Canonical transformations with controlled asymptotic endpoints preserve $\chi$, making it a bridge among PM, amplitude, self-force, and EOB calculations Damour 2020, §§II–III.

When radiation is present, a single conservative angle is insufficient. One must also specify the incoming and outgoing four-momenta, radiated four-momentum, and angular-momentum loss. Comparing a time-symmetric conservative calculation with a retarded numerical evolution without removing dissipative effects mixes distinct observables.

A comparison protocol

Before comparing two results:

1. Match theory, dimension, object types, masses, spins, asymptotics, and boundary conditions.
2. Translate signature, curvature, Fourier, and spin conventions as complete systems.
3. Select common invariant inputs: for example $(\Omega,m_1,m_2)$ or $(E,J,m_1,m_2)$.
4. Express both outputs as the same observable and expand to the same combined order.
5. Check a discriminating limit, such as Newtonian motion, $\nu\to0$, equal masses, zero spin, or zero dissipation.

Agreement in one invariant is necessary but not sufficient for equivalence. Conversely, disagreement in a coordinate potential is not evidence of different physics. The strongest comparisons use several independent invariants because an incorrect transformation can accidentally preserve one relation.

Exercises

Starting from the Schwarzschild circular-geodesic energy above, verify the first two nontrivial terms of the weak-field binding energy.

Solution

Use

$$(1-3x)^{-1/2}=1+\frac32x+\frac{27}{8}x^2+O(x^3).$$

Multiplication by $1-2x$ gives

$$\frac{E_{\mathrm{geo}}}{\mu} =1-\frac12x+\frac38x^2+O(x^3).$$

Subtracting the rest energy yields $e=-x/2+3x^2/8+\cdots$, fixing the test-mass limit of the PN relation.

Two Hamiltonians yield different coordinate radii for the circular orbit but identical $E(\Omega)$ and $K(\Omega)$ through the calculated order. What conclusion is justified?

Solution

The evidence supports equivalence of the tested conservative physics, probably through a coordinate or canonical transformation, provided masses, spins, time normalization, and truncation agree. The two invariant agreements are stronger than agreement in one quantity. They do not prove equivalence for eccentricity, spin, dissipation, or higher orders that were not tested.

References

• Barack, Leor, and Adam Pound. “Self-Force and Radiation Reaction in General Relativity.” Reports on Progress in Physics 82 (2019): 016904. DOI.
• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI.
• Damour, Thibault. “Classical and Quantum Scattering in Post-Minkowskian Gravity.” Physical Review D 102 (2020): 024060. DOI.