Gauge-Invariant Observables and Master Variables

Gauge invariance asks whether a combination of perturbation fields changes when the background-to-physical identification is displaced. Physical observability asks the further question of how that combination is measured, normalized, or reconstructed. This page builds the transformation test, constructs relational and harmonic invariants, and explains why a master variable is powerful without automatically being a local observable.

Helpful background. Black-Hole Geometry and Mechanics fixes the background and horizon data used in applications; PDEs and Green Functions for Gravitation supplies the differential-operator viewpoint; Formulations of General Relativity separates covariant equations, constraints, and evolution; Presymplectic Potentials and Currents prepares the variational language used later in this chapter.

Testing gauge invariance

For any covariantly constructed tensor $Q[g,\psi]$, its first perturbation obeys

$$\delta Q\mapsto \delta Q+\mathcal L_\xi\bar Q.$$

Therefore $\delta Q$ is invariant under the admitted infinitesimal diffeomorphisms precisely when $\mathcal L_\xi\bar Q=0$ for that class of $\xi$. If $\bar Q=0$, the condition holds immediately. It also holds for a constant background scalar and for constant combinations of identity tensors. This is the Stewart–Walker sufficient criterion, stated in its geometric setting and proved as the first-order case of the general gauge-invariance condition in Bruni et al. 1997, arXiv v2, p. 12, Proposition 1, PDF.

The criterion is not the claim that only vanishing background quantities can produce invariants. Several gauge-dependent perturbations may combine so that their Lie-derivative terms cancel. Nor does invariance under small gauge vectors imply invariance under transformations acting at a boundary. An asymptotic time translation, for example, can be a physical symmetry with a charge rather than a redundant identification.

A quick counterexample is the Schwarzschild Kretschmann scalar $\bar K=48G^2M^2/r^6$. Its perturbation transforms as $\delta K\mapsto\delta K+\xi^r\partial_r\bar K$, so $\delta K$ at fixed coordinate $r$ is not gauge invariant. Curvature is geometric, but its value at a coordinate-labeled point is not yet a relational measurement.

Relational combinations of scalar fields

Suppose two background scalars $A$ and $B$ vary only with a parameter $r$ and $d\bar B/dr\ne0$ on the region of interest. Label two gauge representations by $X$ and $Y$. Their perturbations transform as

$$(\delta A)_Y=(\delta A)_X+\xi^r\frac{d\bar A}{dr}, \qquad (\delta B)_Y=(\delta B)_X+\xi^r\frac{d\bar B}{dr}.$$

The combination

$$\delta A_{B} =\delta A-\frac{d\bar A/dr}{d\bar B/dr}\,\delta B$$

is invariant. Operationally it is the change of $A$ on a surface where the value of $B$ is held fixed. It answers a relational question rather than pretending that the background coordinate $r$ is observable.

The construction has a visible failure set. It breaks down where $d\bar B/dr=0$ or where $B$ does not label the desired surfaces uniquely. Angular or temporal components of $\xi$ do not spoil the cancellation: the background gradients of both scalars have only a radial component. Additional scalar markers would be needed to select a unique point within a surface, or a different construction if the backgrounds depend on more coordinates. A different clock or radial marker may define a different, equally legitimate observable. Gauge invariance alone does not erase that physical choice.

For a complete example, compare two Schwarzschild metrics with masses $M$ and $M+\varepsilon\delta M$, using the geometrically defined radius $R=\sqrt{\text{area}/4\pi}$ of their symmetry spheres. In areal coordinates, $\bar R=r$ and $\delta R=0$. After any admissible identification shift,

$$\delta R=\xi^r,\qquad \delta K=\frac{96G^2M\,\delta M}{r^6} +\xi^r\frac{d}{dr}\left(\frac{48G^2M^2}{r^6}\right).$$

Subtraction gives the same answer in both gauges:

$$\delta K_R=\frac{96G^2M\,\delta M}{R^6}.$$

A pure identification change, $\delta M=0$, gives zero. A mass change does not. This example uses spherical families; an arbitrary nonspherical perturbation does not come with a unique family of symmetry spheres whose area automatically defines a pointwise scalar $R$.

Curvature, frames, and boundary observables

On a vacuum type-D background, selected perturbed Weyl scalars are useful because the corresponding background components vanish. Their coordinate gauge transformation therefore vanishes at first order. However, a Weyl component also depends on its null tetrad. One must fix or quotient the tetrad’s boosts, spins, and null rotations before calling the component fully invariant. At infinity, converting curvature to strain additionally requires an asymptotic frame, retarded-time normalization, integrations, and boundary conditions.

Charges provide a complementary class. A perturbation of ADM mass or angular momentum is defined by a surface integral with specified asymptotic falloff and symmetry normalization. Such a quantity can be invariant under gauge vectors that decay sufficiently fast, while changing under an allowed large transformation or a change of asymptotic frame. Local curvature invariants, tetrad components, surface charges, detector observables, and master fields are therefore not interchangeable meanings of “gauge invariant.”

Harmonic invariants and master variables

After decomposing a spherical perturbation into a fixed $(\ell,m)$ and parity sector, collect its amplitudes into $q_A$. Their transformations have the linear form

$$q_A\mapsto q_A+M_A{}^\alpha\xi_\alpha,$$

where $M_A{}^\alpha$ contains background functions and derivatives on the two-dimensional orbit space. A combination $\Psi=C^Aq_A$ is gauge invariant when the differential operator $C^A$ satisfies $C^AM_A{}^\alpha=0$. Constraints can then eliminate nondynamical amplitudes, leaving one master variable per radiative parity sector. Its equation typically takes the form

$$\left(-\partial_t^2+\partial_{r_*}^2-V_\ell(r)\right)\Psi_{\ell m} =S_{\ell m}.$$

The Hamiltonian and Moncrief formulation performs this reduction with the constraints retained explicitly.

An odd-parity cancellation

Here is an explicit invariant before using any field equation. On the unit sphere let $\Omega_{AB}$ be its metric, $D_A$ its derivative, $\epsilon_{\theta\phi}=\sin\theta$, and $Y_{\ell m}$ a unit-normalized scalar harmonic. Define

$$X_A=-\epsilon_A{}^B D_BY_{\ell m}, \qquad X_{AB}=D_{(A}X_{B)},\qquad \ell\ge2.$$

With orbit-space coordinates $x^a=(t,r)$, write the odd components of the metric perturbation as $h_{aB}=h_aX_B$ and $h_{AB}=h_2X_{AB}$. An odd gauge covector $\xi_A=\Lambda(x^a)X_A$, $\xi_a=0$, gives, with this chapter’s active plus convention,

$$h_a'=h_a+r^2D_a(\Lambda/r^2), \qquad h_2'=h_2+2\Lambda.$$

Here $D_a$ is the covariant derivative of the two-dimensional orbit metric. The warp factor $r^2$ comes from the mixed orbit–sphere connection. With $f=1-2GM/r$, the following combinations are invariant:

$$\widetilde h_a=h_a-\frac{r^2}{2}D_a(h_2/r^2), \qquad \Psi_{\mathrm{RW}}=\frac{f}{r}\widetilde h_r$$

The second is written in Schwarzschild coordinates in the open exterior. Indeed the variation of the second term of $\widetilde h_a$ is exactly $r^2D_a(\Lambda/r^2)$. Choosing $\Lambda=-h_2/2$ puts this sector in Regge–Wheeler gauge, where $\widetilde h_a=h_a$. The Regge–Wheeler derivation uses the inverse identification parameter, equal to $-\Lambda$ here, and obtains the same invariant. Its Cunningham–Price–Moncrief field also makes clear why the Regge–Wheeler field alone cannot recover a general stationary sector by division by frequency.

Three qualifications are essential. First, the definition and normalization of $\Psi$ depend on parity, harmonic normalization, and background; formulas from two sources cannot be compared by name alone. Second, the usual radiative master variables assume $\ell\ge2$. The $\ell=0$ and $\ell=1$ sectors contain parameter shifts, center-of-mass or angular-momentum changes, constraints, and gauge pieces that must be restored separately. Third, reconstructing a metric from $\Psi$ requires solving constraints and choosing a gauge; at zeros of a frequency or harmonic factor, a reconstruction formula may fail although the underlying perturbation is regular.

Independent checks and scope limits

An invariant should pass four tests. Substitute the complete gauge transformation and show cancellation without using the field equations unless on-shell invariance is explicitly intended. Check exceptional modes rather than dividing silently by $\ell(\ell+1)-2$. Recover a known flat or Schwarzschild limit in the same normalization. Finally, reconstruct a geometric or flux quantity and verify that two admissible gauges agree.

Exact background construction remains in Exact Solutions of Einstein’s Equations; the background scalars, tetrads, and harmonic geometry used here are inputs. Full finite-amplitude evolution remains in Black-Hole Dynamics. A linear master variable does not by itself describe nonlinear mode coupling, waveform extraction, or the endpoint of a large disturbance.

Exercise: build a relational curvature perturbation

Consider a spherically symmetric family, or another family with a specified covariant choice of two-surfaces. Let $K(r)$ be the background Kretschmann scalar and let $R(r)$ be the resulting areal-radius scalar with $R'\ne0$. Construct the perturbation of $K$ at fixed $R$ and verify its gauge invariance.

Solution

For any identification shift, the radial background dependence gives $\delta K\mapsto\delta K+\xi^rK'$ and $\delta R\mapsto\delta R+\xi^rR'$. Hence

$$\delta K_R =\delta K-\frac{K'}{R'}\delta R$$

transforms into itself because the two terms proportional to $\xi^rK'$ cancel. The result is the change of curvature on a surface of fixed areal radius, not at fixed coordinate label. It is valid only where $R$ is a good local marker; at a critical point of $R$, another relational prescription is needed.

References

• Bruni, Marco, Sabino Matarrese, Silvia Mollerach, and Sebastiano Sonego. “Perturbations of Spacetime: Gauge Transformations and Gauge Invariance at Second Order and Beyond.” Classical and Quantum Gravity 14 (1997): 2585–2606. DOI. Open PDF, arXiv:gr-qc/9609040v2.