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The Noether gauging procedure starts by allowing a constant symmetry parameter to depend on position. For an internal symmetry written as $\delta {\varphi }^i={ϵ}^A{T}_A{\varphi }^i$, integration by parts identifies the Noether current:up to a boundary term. Introduce a gauge field $A_{\mu }^A$ and add $-g{A}_{\mu }^A{j}_A^{\mu }$, with the inhomogeneous gauge transformation $\delta A_{\mu }^A=g^{-1}{\partial }_{\mu }{ϵ}^A+\cdots$. This cancels the displayed variation. The current coupling has its own variation, so one iterates: add the needed higher-order interactions and corrections to the gauge transformations, and require closure of the local symmetry algebra. For an ordinary gauge theory, this reorganizes derivatives into gauge covariant derivatives and fixes the nonlinear gauge field strength. Gauging is therefore more than replacing a constant parameter by a function.

For rigid supersymmetry, the corresponding Noether current is the spinor-valued supercurrent $S^{\mu }$. A local Majorana spinor parameter gives a variation proportional to $({\partial }_{\mu }\overline{ϵ})S^{\mu }$. A gravitino ${\Psi }_{\mu }$, with an inhomogeneous transformation $\delta {\Psi }_{\mu }\sim {\kappa }^{-1}{\partial }_{\mu }ϵ$, cancels it through a coupling proportional to $\kappa {\bar{\Psi }}_{\mu }{S}^{\mu }$. The commutator of two rigid supersymmetry transformations is a translation. With local parameters, the translation parameter becomes position dependent, so the completion must also gauge translations: it introduces the vierbein, diffeomorphism invariance of general relativity, and local Lorentz transformations. The gravitino and graviton thus belong to the same supergravity multiplet.

For pure minimal four-dimensional supergravity, start with the free massless Fierz-Pauli action and the free Rarita-Schwinger field kinetic term. The Noether gauging procedure replaces the former by the Einstein-Hilbert action and couples the latter to the vierbein and spin connection. In a canonical normalization the leading transformations can be writtenAn overall sign or phase changes with the curvature and Majorana spinor conventions, but the relative normalization must be used consistently. To see why these terms fit together, first work to quadratic order in fermions, so the spin connection is torsion free. Varying the Rarita-Schwinger field kinetic term under $\delta {\Psi }_{\mu }={\nabla }_{\mu }ϵ/\kappa$ produces an antisymmetric double spinor covariant derivative. The spinor curvature identity and the Einstein tensor contraction of spin curvature give, in the matching curvature convention,The resulting Einstein tensor term cancels the variation of the Einstein-Hilbert action under $\delta e_{\mu }{}^a$. This is the essential first nontrivial cancellation in the construction.

At higher fermion order, the variations of the vierbein and spin connection produce further terms. Treating the spin connection as independent makes its Euler-Lagrange field equation algebraic. Its solution is a gravitino-induced torsion connection. With ${\Psi }_a=e_a{}^{\mu }{\Psi }_{\mu }$ and ${\gamma }^{ab}=[{\gamma }^a,{\gamma }^b]/2$, a conventional canonical normalization givesEquivalently, with $T^a=d{e}^a+{\tilde{\omega }}^a{}_b\wedge e^b$, the torsion tensor obeysThe Majorana Grassmann bilinear interchange makes the right-hand side antisymmetric in $\mu ,\nu$. Eliminating the spin connection generates the four-fermion completion. In the 1.5-order formalism one substitutes its solution but varies the other fields holding the connection fixed: the omitted chain-rule term is $(\delta S/\delta \omega )\delta \tilde{\omega }=0$. The remaining cubic fermion variations cancel by the Clifford algebra and Fierz rearrangements. This produces the displayed supergravity action in its compact connection-dependent form. Calling this formulation on shell means that no independent auxiliary fields have been retained and that closure of the supersymmetry algebra uses the propagating Euler-Lagrange field equations; it does not mean that action invariance is obtained only by setting every variation to zero on a solution.

In the antisymmetrized kinetic term, ${\Psi }_{\sigma }$ is a spinor-valued one-form and the derivative is conventionally writtenThis is the Lorentz-covariant exterior-derivative convention for that term. If one instead uses the full affine covariant derivative of a vector-spinor, a coordinate-index connection is also present and the antisymmetrization must include its torsion tensor correction. The crucial physical distinction from pure metric general relativity is the spin connection contains gravitino-induced contorsion, rather than being the torsion-free Levi-Civita connection. It reduces to the ordinary spinor covariant derivative when the gravitino vanishes.

Work through terms quadratic in the fermions, with Grassmann-odd Majorana spinors ${\psi }_{\mu }$ and $ϵ$. At this order the spin connection can be taken torsion-free: eliminating its gravitino-induced torsion changes the action only at four-fermion order. The Christoffel term needed to turn the printed $D_{\nu }{\psi }_{\sigma }$ into a full vector-spinor covariant derivative drops out after contraction with ${\gamma }^{\mu \nu \sigma }$, since the connection is symmetric in $\nu ,\sigma$.

First establish the Einstein tensor contraction of spin curvature. Expanding the Clifford product and using the first Bianchi identity, its five-gamma and antisymmetric three-gamma curvature contributions vanish, leavingSince $[{\nabla }_{\nu },{\nabla }_{\rho }]=R_{\nu \rho ab}{\gamma }^{ab}/4$, it follows thatThe factor $1/2$ arises from antisymmetrizing the two derivatives, not from a choice of Einstein-Hilbert normalization.

For the Einstein-Hilbert action, the metric variation isVariation with respect to the covariant metric, including the determinant variation and discarding the usual boundary divergence, givesNo gravitational field equation has been assumed here.

For the Rarita-Schwinger field action, the determinant, gamma-matrix and connection variations each multiply a term already quadratic in $\psi$. Since $\delta e$ is bilinear in $\psi ,ϵ$, these contributions have four fermionic factors and are outside the requested order. Retain only the variations of the two explicit gravitino factors. The four-dimensional Majorana bilinear interchange gives $\bar{\chi }{\gamma }^{\mu \nu \rho }\eta =\bar{\eta }{\gamma }^{\mu \nu \rho }\chi$ for odd spinors. After integration by parts, the first gravitino variation iswhere the last step swaps $\mu ,\rho$ and reverses the three-gamma sign. It is exactly equal to the variation of the second gravitino factor. ThereforeThe curvature contribution cancels $\delta S_{\mathrm{EH}}$ pointwise. Thusup to boundary terms and the omitted four-fermion-order variations. This directly proves the stated local supersymmetry invariance with its printed coefficients. At higher order, the torsion and quartic-fermion completion of minimal four-dimensional supergravity is needed; it cannot be inferred by dropping the terms indefinitely.

Take a complete smooth asymptotically flat spacelike initial-data hypersurface $\Sigma$, carrying a spin structure, without inner boundary. Let $n$ be its future unit normal, $h$ its induced metric, and use the extrinsic curvature convention $k_{ij}=-g({\nabla }_{i}n,e_j)$. Assume the Einstein constraints, the dominant energy condition, and the standard decay sufficient for finite ADM charges. In one asymptotically Cartesian end it is enough here to work with $h_{ij}={\delta }_{ij}+q_{ij}$, $q=O(r^{-1})$, $\partial q=O(r^{-2})$, $k=O(r^{-2})$, and the corresponding differentiated decay and integrability assumptions. The desired conclusion is positivity of the whole ADM four-momentum, not merely of the energy in one selected frame.

Introduce a commuting auxiliary spinor $ϵ$; this is not an anticommuting gravitino field. With $\overline{ϵ}=-{ϵ}^{†}{\gamma }^{\hat{0}}$ its causal Dirac spinor current is $K^{\mu }=\overline{ϵ}{\gamma }^{\mu }ϵ$, future causal. The Nester two-form isIt is real in these conventions and antisymmetric in $\mu ,\nu$. Differentiating, the two terms quadratic in first derivatives become equal after exchanging $\nu ,\rho$. The remaining terms are curvature commutators. The Einstein tensor contraction of spin curvature gives ${\gamma }^{\mu \nu \rho }{\nabla }_{\nu }{\nabla }_{\rho }=G^{\mu }{}_{\lambda }{\gamma }^{\lambda }/2$; the adjoint term supplies the other half. ThusThis is the central spinor-curvature identity, derived by differentiation rather than posited as a positivity formula.

Project onto $\Sigma$ and put $b^i=n_{\mu }{B}^{\mu i}$. In an adapted orthonormal frame $n_{\mu }=(-1,0,0,0)$, Clifford algebra givesIndeed the expansion of $|{\gamma }^i{\nabla }_iϵ{|}^2$ is ${\sum }_i|{\nabla }_iϵ{|}^2+({\nabla }_iϵ{)}^{†}{\gamma }^{ij}{\nabla }_jϵ$. With a Gaussian normal extension of the slice, the symmetric second fundamental form has zero contraction with the antisymmetric $B$, so the projected divergence is the intrinsic spatial divergence. This proves the Witten-Nester identityHere $D_i$ on $b$ denotes the spatial metric derivative. On spinors, ${\nabla }_i$ is the Sen spinor connection, the restriction of the spacetime connection, not just the intrinsic three-dimensional one:The extrinsic-curvature term is essential when the data have momentum.

Choose $ϵ$ to solve the Witten spinor equation, $\mathcal{D}ϵ=0$, approaching a prescribed constant ${ϵ}_{\infty }$ in the chosen end and zero constants in any other ends. The Einstein constraints identify $G_{\mu \nu }{n}^{\mu }{K}^{\nu }=8\pi G{T}_{\mu \nu }{n}^{\mu }{K}^{\nu }$. The dominant energy condition makes this nonnegative because both $n$ and $K$ are future causal. Consequently the right-hand side of the identity becomes a sum of nonnegative terms.

Existence of that spinor is an analytic part of the proof, not something established by the positivity identity alone. Extend ${ϵ}_{\infty }$ smoothly using a cutoff and solve $\mathcal{D}u=-\mathcal{D}{ϵ}_0$ for a decaying correction. The spatial principal symbol ${\gamma }^i{\xi }_i$ squares to $|\xi {|}_h^2$, so the equation is elliptic. For a compactly supported spinor, the integrated identity givesTogether with the weighted asymptotic estimates this gives the coercivity needed for elliptic existence of a Witten spinor. A decaying homogeneous solution would be parallel by the same identity and hence zero, excluding the kernel; the appropriate weighted elliptic solvability gives the correction. Under the stated smooth decay assumptions it has $ϵ-{ϵ}_{\infty }=O(r^{-1})$ and first derivative $O(r^{-2})$. The rigorous analytic step is supplied by the spinor existence proof. This also explains why completeness, asymptotic data and the absence of uncontrolled inner boundary terms are hypotheses of the argument.

It remains to identify the boundary integral with the physical ADM charge. In the asymptotic symmetric coframe $e_i{}^a={\delta }_i{}^a+q_{ia}/2+O(r^{-2})$, the leading spatial spin connection isInsert this and the Sen correction into $b^i$. The real scalar part of ${\gamma }^{ij}{\gamma }^{kl}$ is ${\delta }^{il}{\delta }^{jk}-{\delta }^{ik}{\delta }^{jl}$; its two-gamma part has imaginary expectation and drops out. It follows that the intrinsic connection contributes $\frac{1}{2}({\partial }_j{q}_{ij}-{\partial }_i{q}_{jj})K_{\infty }^{\hat{0}}$. For the other term, symmetry of $k$ removes the three-spatial-gamma product, leaving $-(k_{ij}-k{h}_{ij})K_{\infty }^{\hat{j}}$. Thus the ADM boundary term of the Nester two-form has the expansionThe constant-spinor derivative contribution integrates to zero on a large coordinate sphere. Its coefficient is antisymmetric in $i,j$; the corresponding tangential vector on the sphere is divergence free, so integration by parts removes it. Equivalently, the flux of ${\gamma }^{ij}{\partial }_{j}u$ has identically zero divergence, with its asymptotic leading term having zero flux. The quadratic correction terms are $O(r^{-3})$ and their surface integrals vanish in the limit.

Define the ADM energy and ADM momentum with the same conventions:The boundary calculation then gives, including every normalization factor,This is the energy associated with the asymptotic spinor's causal translation. In the zero-momentum rest frame it is the ADM mass times $K_{\infty }^{\hat{0}}$; more generally it proves the full energy-momentum bound.

Finally apply the divergence theorem to the Witten identity. There is no inner boundary, and the spinors assigned zero at other ends give no charge there. ThereforeAsymptotic constant spinors can produce every future null direction. Normalize $K_{\infty }^{\hat{0}}=1$ and choose its spatial direction along $\mathbf{P}$; the inequality givesThis establishes the positive energy theorem and nonnegativity of the invariant mass when it is defined. For zero ADM energy, repeat the identity for a basis of asymptotic spinors: all become parallel and all matter contractions vanish. Their curvature integrability forces the spacetime curvature to vanish on the initial surface; the standard rigidity conclusion is that the data embed in Minkowski spacetime. Merely having a parallel null spinor in a local wave geometry would not by itself prove this global rigidity: completeness and asymptotic flatness remain essential.

In mostly-plus signature orient the commuting Dirac current to the future by $\overline{ϵ}=-{ϵ}^{†}{\gamma }^{\hat{0}}$. For the Nester two-form, put $b^i=n_{\mu }{B}^{\mu i}$. Its covariant divergence has a spinor-gradient term and the Einstein tensor contraction of spin curvature. Projecting onto the hypersurface gives the displayed identity with positive spinor norm. The Witten spinor equation removes the negative square, and the dominant energy condition makes the matter term nonnegative after the Einstein constraints are imposed.