Actions, Fields, and Symmetries

A gravitational theory is specified by more than a formula for its bulk field equations. One must also say which fields are varied, which transformations are gauge, how matter identifies physical rods and clocks, which surface terms and data define the variational problem, and which solutions are admitted. These ingredients decide whether two actions describe the same physics or merely look similar. They also determine the constraint algebra, charges, and observables used throughout this volume.

Required background. Einstein Equations supplies the geometric and matter sides of the GR baseline; Gravitational Actions and Boundary Terms supplies the boundary completion of a gravitational variational principle; Variational and Symplectic Geometry supplies the field-space variation, gauge degeneracy, and charge language used below.

Helpful background. Formulations of General Relativity supplies the constraint, gauge-fixing, and initial-value distinctions used to interpret the theory specification.

From variables to a variational problem

On a $D$-dimensional spacetime $M$, collect all dynamical fields into $\Phi^A$. The label $A$ includes tensor, internal, and species indices. A local theory with finitely many derivatives has a Lagrangian $D$-form $\mathbf L[\Phi]$ whose first variation can be arranged as

$$\delta\mathbf L =\mathbf E_A(\Phi)\,\delta\Phi^A +d\boldsymbol\Theta(\Phi,\delta\Phi).$$

The Euler–Lagrange equations are $\mathbf E_A=0$; the $(D-1)$-form $\boldsymbol\Theta$ is the presymplectic potential current. Integrations by parts fix this decomposition only up to exact terms, but they do not make the surface contribution optional. The action

$$S[\Phi]=\int_M\mathbf L+\int_{\partial M}\mathbf B$$

defines a differentiable variational problem only when the boundary completion $\mathbf B$ and the allowed variations make $\boldsymbol\Theta+\delta\mathbf B$ vanish, or reduce it to the intended boundary equations. For the Einstein–Hilbert density, fixing the induced metric on a non-null boundary requires the Gibbons–Hawking–York term. Omitting it leaves normal derivatives of $\delta g_{ab}$ in the surface variation even though the same vacuum Einstein equation appears in the interior. Thus equal bulk equations need not mean equal phase spaces, Hamiltonians, or charges Gibbons and Hawking 1977, pp. 2752–2754.

A compact, reusable specification is

$$\mathfrak T =\bigl(M,\Phi,\mathcal G,S,\mathcal B,\mathcal O\bigr),$$

with the following entries.

Entry	What must be fixed	Why it changes the physics
$M$	Dimension, differentiable and causal structure, topology, orientation, asymptotic regions, and any spin or bundle structure	Local equations do not determine global sectors, zero modes, or allowed charges.
$\Phi$	Dynamical fields, nondynamical structures, reality and regularity conditions, and which connection or frame is independent	Varying a connection is different from declaring it Levi–Civita; a reference metric is not a second dynamical metric.
$\mathcal G$	Gauge transformations, reducibility, exact global symmetries, and allowed large transformations	Gauge directions are quotiented from phase space, whereas global symmetries can carry charges or label sectors.
$S$	Bulk operators and coefficients, matter action, boundary and corner terms, and the order retained in an EFT	These data determine the equations, source couplings, symplectic structure, and approximation error.
$\mathcal B$	Initial and boundary data, horizon or center regularity, falloff, Green-function prescription, and branch choice	The same differential equations can define inequivalent causal or boundary-value problems.
$\mathcal O$	Matter metric, relational or asymptotic observables, units, and equivalence relation on solutions	A change of variables preserves physics only if the quantities actually measured are transformed with it.

The tuple is deliberately redundant enough to prevent a common shortcut: presenting an action and silently importing the rest from GR. An extra scalar, vector, connection, or metric changes the admissible data and physical modes even if a familiar metric remains among the variables.

Gauge symmetry, identities, and charges

Consider an infinitesimal diffeomorphism generated by $\xi^a$. Covariant fields transform actively as $\delta_\xi\Phi^A=\mathcal L_\xi\Phi^A$, while a covariant Lagrangian form obeys $\delta_\xi\mathbf L=\mathcal L_\xi\mathbf L=d(\xi\mathbin{\cdot}\mathbf L)$. Define the Noether current

$$\mathbf J_\xi =\boldsymbol\Theta(\Phi,\mathcal L_\xi\Phi) -\xi\mathbin{\cdot}\mathbf L.$$

Combining the two variations gives the off-shell identity

$$d\mathbf J_\xi =-\mathbf E_A\,\mathcal L_\xi\Phi^A.$$

Because $\xi^a$ and its derivatives can be chosen locally, integrating the right-hand side by parts yields differential identities among the $\mathbf E_A$. In metric GR the result is the contracted Bianchi identity, which makes $\nabla_aT^{ab}=0$ compatible with the Einstein equation. On shell, $d\mathbf J_\xi=0$ and locally $\mathbf J_\xi=d\mathbf Q_\xi$ up to constraint terms; surface integrals of the Noether charge $(D-2)$-form $\mathbf Q_\xi$ underlie asymptotic charges and black-hole mechanics. The derivation, including its ambiguity structure, is given in Iyer and Wald 1994, § 4, equations (49)–(59), PDF.

This calculation separates three notions that are often conflated.

• A gauge transformation connects descriptions of one physical state when its parameter is an allowed degeneracy direction. It generates constraints, not a new conserved particle number.
• A large transformation or asymptotic symmetry need not be a degeneracy; its surface term can carry a charge. Which parameters are allowed depends on the falloff and boundary conditions.
• A global symmetry acts nontrivially on physical states and has a current unrelated to quotienting coordinate descriptions.

The distinction cannot be read from the symbol used for a transformation. One must inspect the action, presymplectic form, allowed parameters, and surface terms. Lee and Wald formulate this relation between local symmetries, constraints, and reduced phase space in Lee and Wald 1990, pp. 726–732.

Matter coupling fixes the measuring geometry

Suppose the gravitational variables are $(g_{ab},\phi)$, but matter appears as

$$S_m[\widetilde g,\psi], \qquad \widetilde g_{ab}=A^2(\phi)g_{ab}.$$

Freely falling matter, clock rates, stress tensors, and measured masses are defined with $\widetilde g_{ab}$, not automatically with $g_{ab}$. An invertible conformal change can rewrite the same classical theory in variables where the gravitational kinetic terms look different, provided the matter action, sources, boundary data, and units are transformed too. Holding the matter action fixed while changing only the gravitational part instead changes the tuple $\mathfrak T$ and generally changes observables. The next page makes this criterion explicit for Jordan, Einstein, and disformal frames.

Nondynamical structures require the same care. In massive gravity a reference metric is prescribed data; in bimetric gravity both metrics are varied. In metric–affine gravity the connection is independent; in metric GR it is determined by the metric. In a nonlocal model the notation $\Box^{-1}$ is incomplete until a Green function and homogeneous data are specified. These are differences in theory definition, not cosmetic conventions.

Two independent checks

The variational identity can be checked without gravity. For Maxwell theory,

$$S[A]=-\frac14\int_M d^4x\sqrt{-g}\,F_{ab}F^{ab}, \qquad F_{ab}=2\nabla_{[a}A_{b]},$$

integration by parts gives

$$\delta S =\int_M d^4x\sqrt{-g}\,(\nabla_aF^{ab})\,\delta A_b -\int_{\partial M}d\Sigma_a\,F^{ab}\delta A_b.$$

Thus $E^b=\nabla_aF^{ab}$ and $\Theta^a=-F^{ab}\delta A_b$. Under $\delta_\lambda A_b=\nabla_b\lambda$, gauge invariance implies $\nabla_bE^b=0$, which also follows directly from antisymmetry of $F^{ab}$. One can make the variational principle well posed by fixing the pullback of $A_b$ at the boundary, or by adding the appropriate Legendre surface term and fixing electric flux instead. The bulk Maxwell equation is unchanged, but the admissible boundary phase space is not.

A second check is dimensional and structural. In four dimensions $[G]=-2$, $[R]=2$, and $d^4x\sqrt{-g}R/G$ is dimensionless in natural units. Every term in $\delta\mathbf L$ therefore has the same mass dimension as $\mathbf L$, while $\boldsymbol\Theta$ has one less form degree but not an arbitrary normalization. Setting the surface term to zero before declaring the boundary data fails this check: it removes information that is present in the variation rather than deriving a condition from the theory.

Common pitfalls

“Same equations” means “same theory.” Bulk equations can coincide while boundary conditions, global sectors, symplectic forms, matter couplings, or observables differ. Equivalence must be established for the full tuple $\mathfrak T$ in the stated domain.

Every symmetry is a gauge redundancy. Gauge transformations with vanishing surface charge are redundant descriptions. Large or asymptotic transformations may act on physical states and carry charges, while global symmetries are not quotiented at all.

The field list is the mode count. Lapse, shift, gauge potentials, and other components may impose constraints rather than propagate. Physical degrees of freedom are counted only after the complete constraint algorithm and boundary qualification developed in the next chapter.

Exercises

For Maxwell theory on a finite timelike region, derive the Euler–Lagrange term and presymplectic potential above. Then show the gauge Noether identity and compare Dirichlet potential data with fixed electric-flux data.

Solution

Using $\delta F_{ab}=2\nabla_{[a}\delta A_{b]}$ and antisymmetry,

$$-\frac12F^{ab}\delta F_{ab} =-F^{ab}\nabla_a\delta A_b =(\nabla_aF^{ab})\delta A_b -\nabla_a(F^{ab}\delta A_b).$$

After integration, the first term gives $E^b=\nabla_aF^{ab}$ and the second gives $\Theta^a=-F^{ab}\delta A_b$. For a gauge variation,

$$E^b\nabla_b\lambda =\nabla_b(E^b\lambda)-\lambda\nabla_bE^b.$$

Gauge invariance for arbitrary interior $\lambda$ therefore requires $\nabla_bE^b=0$. Directly, $\nabla_b\nabla_aF^{ab}=\tfrac12[\nabla_b,\nabla_a]F^{ab}=0$: the two Ricci contractions cancel because the Ricci tensor is symmetric and $F^{ab}$ is antisymmetric.

For Dirichlet data, require the tangential pullback of $\delta A_b$ to vanish; then the boundary variation vanishes. To exchange potential for normal electric flux, add

$$S_{\mathrm L}=\int_{\partial M}d\Sigma_a\,F^{ab}A_b.$$

Its variation cancels the $F^{ab}\delta A_b$ term and leaves, up to the fixed boundary geometry, $A_b\,\delta(n_aF^{ab})$. Fixing $n_aF^{ab}$ then gives a differentiable action. Gauge parameters at the boundary must be restricted or accompanied by the corresponding boundary charge; otherwise the transformation changes the prescribed data. The two problems share the bulk Maxwell equation but not the same boundary ensemble or canonical generators.

References

• Gibbons, G. W., and S. W. Hawking (1977), “Action Integrals and Partition Functions in Quantum Gravity,” Physical Review D 15, 2752–2756. DOI.
• Iyer, Vivek, and Robert M. Wald (1994), “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy,” Physical Review D 50, 846–864. DOI; Open PDF.
• Lee, Joohan, and Robert M. Wald (1990), “Local Symmetries and Constraints,” Journal of Mathematical Physics 31, 725–743. DOI.