Gauge Transformations in General Relativity

A change in metric components is only a change of representation when it comes from a diffeomorphism applied consistently to the metric, matter fields, initial hypersurface, and admissible boundary data. The same statement has finite, infinitesimal, coordinate, and initial-value forms. Keeping those forms together is essential: a component change can be pure gauge in the interior yet act as a physical symmetry at a boundary, and two constraint-satisfying initial data sets need not be gauge equivalent merely because they look similar.

Required background. The Einstein Equation supplies the covariant field equation, while Killing Fields and Isometry Groups supplies flows, pullbacks, Lie derivatives, and exact symmetries.

Helpful background. Hypersurface Normals and Tangential Projectors supplies induced initial data, and PDEs and Green Functions for Gravitation supplies the initial- and boundary-value language used to restrict gauge transformations.

Finite transformations of the Einstein fields

Let a field configuration on a manifold $\mathcal M$ be

$$\mathcal F=(g_{\mu\nu},\psi),$$

where $\psi$ denotes all matter fields. For a diffeomorphism $\Phi:\mathcal M\to\mathcal M$, adopt the active pullback convention

$$\mathcal F\longmapsto\Phi^*\mathcal F =(\Phi^*g,\Phi^*\psi).$$

Naturality of the Einstein tensor gives

$$G[\Phi^*g]+\Lambda\Phi^*g =\Phi^*\!\left(G[g]+\Lambda g\right).$$

If the matter equations are covariant and $G[g]+\Lambda g=8\pi G T[g,\psi]$, then $(\Phi^*g,\Phi^*\psi)$ is another solution. The transformation moves every field together. It does not compare $\Phi^*g$ with an untransformed material clock or detector.

A passive coordinate change describes the same operation with a different question held fixed. If $x^\mu$ and $x^{\mu'}$ label the same event, then

$$g_{\mu'\nu'}(x') =\frac{\partial x^\mu}{\partial x^{\mu'}} \frac{\partial x^\nu}{\partial x^{\nu'}} g_{\mu\nu}(x(x')).$$

The geometric metric is unchanged; its basis components and their functional form change. Under the active map, by contrast, $\Phi^*g$ is compared with $g$ at the same labeled point. Carroll develops this active–passive distinction and the pullback construction in Carroll 1997, §5, pp. 129–141, PDF.

These descriptions encode the same physics only after the comparison map is declared. For example, a scalar coincidence such as “the curvature has value $R_0$ where a clock reads $\tau_0$” is carried to the corresponding coincidence under $\Phi$. Comparing $R(p)$ with $(\Phi^*R)(p)=R(\Phi(p))$ while refusing to move the clock is a different physical comparison.

The pages on active and passive diffeomorphisms and diffeomorphism gauge freedom give the covariant physical interpretation. Here the additional question is which transformations are admissible for an initial or boundary formulation.

Infinitesimal gauge directions

Let $\Phi_s$ be the flow of a vector field $\xi^\mu$, with $\Phi_0$ the identity. Differentiating the chosen pullback action fixes the sign:

$$\left.\frac{d}{ds}\Phi_s^*g_{\mu\nu}\right|_{s=0} =\mathcal L_\xi g_{\mu\nu} =2\nabla_{(\mu}\xi_{\nu)},$$

and, for a scalar matter field,

$$\left.\frac{d}{ds}\Phi_s^*\varphi\right|_{s=0} =\mathcal L_\xi\varphi =\xi^\mu\nabla_\mu\varphi.$$

There is no invariant dispute about an overall sign. A passive change $x^{\mu'}=x^\mu+s\xi^\mu$ produces a component variation with the opposite sign at fixed numerical argument, while using the inverse flow for the active map does the same. A calculation must choose one map and retain it throughout.

For perturbations, write

$$g_{\mu\nu}=\bar g_{\mu\nu}+\varepsilon h_{\mu\nu}+O(\varepsilon^2)$$

and take a diffeomorphism generated by $\varepsilon\xi^\mu$. Holding the chosen background identification fixed gives

$$h_{\mu\nu}\longmapsto h_{\mu\nu}+\mathcal L_\xi\bar g_{\mu\nu} =h_{\mu\nu}+2\bar\nabla_{(\mu}\xi_{\nu)}.$$

This is a first-order change of the identification between the physical and background manifolds, not a second physical perturbation. On a flat background, substitution into the linearized Riemann tensor makes all third derivatives of $\xi$ cancel pairwise. Thus a nonzero $h_{\mu\nu}$ can be pure gauge even though its components vary in space and time. The converse is local: vanishing linearized curvature does not by itself remove global identifications or boundary data.

An exact Killing field satisfies $\mathcal L_\xi g=0$. It belongs to the stabilizer of that solution: its flow changes neither the metric nor its components in an adapted description. This is distinct from a generic gauge direction, which moves along an orbit of different representatives.

Gauge-equivalent initial data

On a spacelike hypersurface $\Sigma$, geometric initial data consist of an induced metric $\gamma_{ij}$, extrinsic curvature $K_{ij}$, and appropriate matter data, all satisfying the constraint equations. A spatial diffeomorphism $f:\Sigma\to\Sigma$ acts by

$$(\gamma,K,\psi_\Sigma,\Pi_\Sigma) \longmapsto (f^*\gamma,f^*K,f^*\psi_\Sigma,f^*\Pi_\Sigma).$$

Because the constraints are natural tensor equations, the pulled-back data satisfy them whenever the original data do. If $f$ preserves the allowed asymptotic end, inner boundary, orientation, and matter reference data, the two sets describe the same initial geometry with relabeled points.

A spacetime diffeomorphism can also change the embedding of $\Sigma$. Its tangential part relabels points within the slice; its normal part changes which slice of the same spacetime development is used as initial data. The lapse and shift describe this choice of evolution vector and are not additional intrinsic gravitational data. Gourgoulhon separates adapted coordinates from the geometric Cauchy data in Gourgoulhon 2007, §§4.2–4.4, pp. 54–64, PDF.

This yields a precise test. Two data sets are gauge equivalent only if an admissible diffeomorphism maps all their geometric and matter data, including the embeddings and boundaries where relevant. Equality of a few scalar invariants is insufficient, and satisfying the same constraints is far too weak: the constraint surface contains physically inequivalent initial states.

Boundaries decide what is gauge

The field equations alone map every diffeomorphism-related solution to another solution. A formulated problem contains more structure: a domain, initial surface, boundary conditions, asymptotic falloff, and sometimes external reference fields. Those structures determine which solution maps are redundancies.

Transformation	What it preserves	Status in the formulated problem
Passive coordinate change	Every geometric field and event	Always a relabeling
Active map supported away from prescribed initial and boundary surfaces	Fixed initial and boundary data	Gauge redundancy
Spatial pullback preserving an asymptotic end	Initial geometry and admissible falloff	Gauge, subject to the stated asymptotic frame
Map nontrivial at a boundary	Bulk equations but possibly boundary fields	May be a boundary symmetry with charge
Map that changes prescribed boundary data	A different boundary-value problem	Not gauge within the original problem
Pulling back only the metric	Not the complete matter–geometry configuration	Generally a physical comparison

A large diffeomorphism may preserve local equations and falloff conditions yet change global or asymptotic data. Conversely, a vector field that vanishes with the required derivatives at the boundary is normally a degenerate direction of the physical description. Later chapters make this distinction quantitative through boundary Hamiltonians and the covariant presymplectic form.

Topology supplies another limit: a diffeomorphism is an invertible smooth map of one manifold. It cannot turn one topology into another or repair a singular, noninvertible coordinate map. A coordinate chart whose Jacobian vanishes has ceased to define a diffeomorphism; invoking “gauge freedom” does not make the chart admissible.

A flat-spacetime check

Start with two-dimensional Minkowski spacetime in coordinates $(T,X)$ and set

$$T=t, \qquad X=x+F(t).$$

Where $F$ is smooth, the Jacobian is one and the transformation is a diffeomorphism. The line element becomes

$$ds^2 =-dt^2+\bigl(dx+\dot F\,dt\bigr)^2 =-\bigl(1-\dot F^2\bigr)dt^2 +2\dot F\,dt\,dx+dx^2.$$

The components are time dependent when $\dot F$ is, but the Riemann tensor is still zero. In lapse–shift language the slices are the same inertial $T=\text{constant}$ lines, with lapse $\alpha=1$, spatial metric $\gamma_{xx}=1$, shift $\beta^x=\dot F$, and $K_{xx}=0$. The shift records the motion of coordinate labels along a fixed slice; it has not created gravity.

This example performs three independent checks at once: pullback naturality keeps curvature zero, the intrinsic initial data are unchanged, and a proper distance computed between consistently mapped endpoints is the same. If only the metric is transformed while the endpoints are held as external labels, the last comparison changes—exactly as the relational argument predicts.

Common pitfalls

Covariance is not complete gauge fixing. Covariant equations map gauge orbits to themselves; they do not choose a unique representative. Differential coordinate conditions still admit residual solutions determined by initial and boundary data.

A scalar is not automatically an observable. A scalar has a coordinate-independent value at a specified event, but identifying “the same event” across gauge-related solutions requires relational or boundary data.

Constraint satisfaction is not gauge equivalence. The constraints select admissible initial states. They do not collapse all such states to one gauge orbit; distinct gravitational radiation or matter data can satisfy the same constraint equations.

Exercise: a moving spatial coordinate

For the flat-spacetime transformation $T=t$, $X=x+F(t)$ above:

1. read off the ADM lapse, shift, and spatial metric;
2. show directly that the induced metric and extrinsic curvature of each $t=\text{constant}$ slice agree with those in $(T,X)$ coordinates; and
3. explain why a boundary fixed at $x=x_0$ is not the same physical worldline as a boundary fixed at $X=x_0$ unless $F=0$ throughout the time interval.

Solution

Comparing

$$ds^2=-\alpha^2dt^2 +\gamma_{xx}(dx+\beta^xdt)^2$$

with the transformed line element gives $\alpha=1$, $\beta^x=\dot F$, and $\gamma_{xx}=1$. A constant-$t$ slice is also a constant-$T$ slice, so its induced line element is $dx^2=dX^2$ at fixed time. Its unit normal is the inertial vector $\partial_T$; equivalently, $n=\partial_t-\dot F\,\partial_x$. Since the spatial metric is constant and the shift Lie derivative of $\gamma_{xx}$ vanishes, the metric-evolution formula gives $K_{xx}=0$.

The curve $x=x_0$ has $X=x_0+F(t)$, whereas $X=x_0$ is inertial and has $x=x_0-F(t)$. They coincide for these same numerical labels only if $F=0$. A nonzero constant $F$ gives two parallel inertial worldlines; one can match them only by shifting the boundary label consistently. A time-dependent $F$ can also change the boundary motion. Fixing one or the other as a boundary therefore generally supplies different geometric boundary data. The coordinate transformation is gauge only when the boundary is transformed with the fields.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.
• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.