Infinitesimal Diffeomorphisms and Gauge Transformations

Perturbation theory compares fields on different spacetimes, so it must say which point of the background represents each point of the physical spacetime. Changing that identification is a gauge transformation. This page derives the first- and second-order laws with one explicit sign convention, then separates this freedom from passive coordinate relabeling and from a physical change of the solution.

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.

Identification maps between spacetimes

Let $\mathcal M_\varepsilon$ carry the physical tensor $Q_\varepsilon$, and let a choice of maps $X_\varepsilon:\mathcal M_0\to\mathcal M_\varepsilon$ pull it back to one background manifold. In the gauge $X$,

$$X_\varepsilon^*Q_\varepsilon =Q^{(0)}+\varepsilon Q_X^{(1)} +\frac12\varepsilon^2Q_X^{(2)}+O(\varepsilon^3).$$

A second choice $Y_\varepsilon$ gives a diffeomorphism of the background, $\Phi_\varepsilon=X_\varepsilon^{-1}\circ Y_\varepsilon$. We choose its generators so that, when acting by pullback,

$$\Phi_\varepsilon^* =1+\varepsilon\mathcal L_{\xi_1} +\frac12\varepsilon^2 \left(\mathcal L_{\xi_2}+\mathcal L_{\xi_1}^2\right) +O(\varepsilon^3).$$

The fields in gauge $Y$ are $Y_\varepsilon^*Q_\varepsilon=\Phi_\varepsilon^*X_\varepsilon^*Q_\varepsilon$. Multiplying the two Taylor series gives the transformation rules

$$Q_Y^{(1)}=Q_X^{(1)}+\mathcal L_{\xi_1}Q^{(0)},$$ $$Q_Y^{(2)}=Q_X^{(2)} +\mathcal L_{\xi_2}Q^{(0)} +2\mathcal L_{\xi_1}Q_X^{(1)} +\mathcal L_{\xi_1}^2Q^{(0)}.$$

The factor of two in the mixed term follows from the $1/2$ in the second-order field expansion. These formulas, including the fact that a general one-parameter family need not be a group, are derived in Bruni et al. 1997, arXiv v2, p. 6, Eq. (2.13), and pp. 13–14, Eqs. (4.4)–(4.12), PDF. Only sufficient differentiability for this finite Taylor expansion is needed; convergence of an infinite analytic series is not being assumed.

Metric and matter transformations

For the metric family $g=\bar g+\varepsilon h+\tfrac12\varepsilon^2 k+\cdots$, the general laws specialize to

$$h'_{\mu\nu} =h_{\mu\nu}+2\bar\nabla_{(\mu}\xi_{1\nu)},$$ $$k'_{\mu\nu} =k_{\mu\nu} +\mathcal L_{\xi_2}\bar g_{\mu\nu} +2\mathcal L_{\xi_1}h_{\mu\nu} +\mathcal L_{\xi_1}^2\bar g_{\mu\nu}.$$

Parentheses denote weight-one-half symmetrization. A scalar matter field obeys $\delta\psi'=\delta\psi+\xi_1^\mu\bar\nabla_\mu\bar\psi$ at first order, and the same tensorial rule applies to vectors, forms, and stress–energy. Metric and matter transformations cannot be chosen independently: a diffeomorphism moves every background field.

Our plus signs define the direction of $\Phi_\varepsilon$. Authors who compare the inverse identification write minus signs and use generators $-\xi_1,-\xi_2$ with the appropriate second-order composition. Apparent disagreement disappears only after translating both the map direction and the $1/2$ convention. The invariant check is covariance: $E^{(1)}[\mathcal L_{\xi_1}\bar g, \mathcal L_{\xi_1}\bar\psi]=\mathcal L_{\xi_1}\bar E$, which vanishes on a background solution.

Active and passive transformations

A passive coordinate transformation changes component labels while leaving the geometric tensor fixed. An active diffeomorphism pulls the tensor field along a vector field. A change of perturbative identification uses an active diffeomorphism on the background to compare two pullbacks. These descriptions can yield the same component formula, but they answer different questions.

A physical perturbation instead changes the diffeomorphism class of the solution or its relational observables. For example, an infinitesimal change of Kerr mass is not generally pure gauge under transformations required to vanish at infinity: it changes the asymptotic charge. Conversely, a field $h=\mathcal L_\xi\bar g$ with an admissible $\xi$ is a gauge direction even when its components look large near a coordinate singularity. Boundary behavior matters; a vector field that acts nontrivially at infinity or at a boundary may generate a symmetry with a charge rather than a degeneracy.

The first-order Stewart–Walker criterion gives a useful sufficient test. If a covariantly defined background tensor vanishes, is a constant scalar, or is a constant combination of identity tensors, its first perturbation is unchanged by arbitrary $\xi_1$. The converse requires care about the class of admitted generators. Bruni and collaborators state the all-order criterion and prove that its first-order case is exactly $\mathcal L_\xi Q^{(0)}=0$ for every admitted generator (Bruni et al. 1997, arXiv v2, p. 12, Proposition 1, PDF).

Second-order composition and consistency

Second order is not obtained by applying the linear rule twice while holding $h$ fixed. The term $2\mathcal L_{\xi_1}h$ says that the first-order field itself is moved; $\mathcal L_{\xi_1}^2\bar g$ records two successive first-order moves; $\xi_2$ contains genuinely second-order information about the family of maps.

Write $T_\xi=\Phi_\xi^*$ for the displayed pullback series, with generator pair $(\xi_1,\xi_2)$, and similarly for $T_\eta$. In the ordered product $T_\eta T_\xi$, the $\xi$ transformation acts on a field first. Multiplication gives

$$\begin{aligned} T_\eta T_\xi ={}&1+\varepsilon(\mathcal L_{\xi_1}+\mathcal L_{\eta_1})\\ &+\frac{\varepsilon^2}{2}\left( \mathcal L_{\xi_2}+\mathcal L_{\eta_2} +\mathcal L_{\xi_1}^2+\mathcal L_{\eta_1}^2 +2\mathcal L_{\eta_1}\mathcal L_{\xi_1}\right) +O(\varepsilon^3). \end{aligned}$$

Using $[\mathcal L_\eta,\mathcal L_\xi]=\mathcal L_{[\eta,\xi]}$, this is a single pullback with effective generators

$$\zeta_1=\xi_1+\eta_1,\qquad \zeta_2=\xi_2+\eta_2+[\eta_1,\xi_1].$$

The order is essential: $T_\eta T_\xi=(\Phi_\xi\circ\Phi_\eta)^*$, because pullbacks reverse map composition. Interchanging the two operators reverses the bracket term.

For a direct check on the plane, take $\xi_1=x\partial_y$, $\eta_1=y\partial_x$, and $\xi_2=\eta_2=0$. Then $[\eta_1,\xi_1]=-x\partial_x+y\partial_y$. Acting on the scalar coordinate function $x$, first $T_\xi x=x$, then $T_\eta x=x+\varepsilon y$. In the effective series, $\mathcal L_{\zeta_1}^2x=x$ whereas $\mathcal L_{\zeta_2}x=-x$: the second-order terms cancel exactly. The opposite bracket sign would incorrectly leave an $\varepsilon^2x$ term. A second-order gauge choice must therefore specify the composition order, not only its first-order vector.

Exact background construction remains in Exact Solutions of Einstein’s Equations: the identification maps compare an already selected family and do not create that family. Full finite-amplitude evolution remains in Black-Hole Dynamics. A second-order transformation law controls the truncated expansion, but it does not guarantee that the expansion is uniform at late times or near a singular limit.

Exercise: translate the opposite sign convention

Suppose another source defines the comparison map as $\widetilde\Phi_\varepsilon=\Phi_\varepsilon^{-1}$ and writes $\widetilde h=h-\mathcal L_\zeta\bar g$ at first order. Find $\zeta$ in terms of $\xi_1$ and explain why inserting only that relation is insufficient at second order.

Solution

To first order, $\Phi_\varepsilon^{-1*}=1-\varepsilon\mathcal L_{\xi_1}+O(\varepsilon^2)$, so $\zeta=\xi_1$ in the source’s displayed minus-sign convention (equivalently its signed generator is $-\xi_1$ relative to ours). Multiplying the original series by its inverse gives

$$T_\xi^{-1} =1-\varepsilon\mathcal L_{\xi_1} +\frac{\varepsilon^2}{2} \left(\mathcal L_{\xi_1}^2-\mathcal L_{\xi_2}\right) +O(\varepsilon^3).$$

For this single family the inverse signed generator pair is $(-\xi_1,-\xi_2)$; the squared Lie derivative retains a plus sign. One must also reverse the order when inverting a product of maps, which changes the sign of the relevant Lie bracket. Therefore replacing $\xi_1$ by $-\xi_1$ in the linear formula alone cannot translate the full second-order rule; the definitions of $\xi_2$ and the $1/2$ normalization must be translated as well.

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.