Noether's Theorem: Symmetries to Memory Effects

Noether's first theorem applies to global (rigid) symmetries. Given an action $S[\phi]$ invariant under a continuous transformation $\phi \to \phi + \epsilon\,\delta\phi$, the on-shell conservation law is:

where $K^\mu$ accounts for the case where the Lagrangian transforms by a total divergence: $\delta\mathcal{L} = \partial_\mu K^\mu$. The conserved charge is:

For a scalar field with time-translation invariance, $j^0 = \mathcal{H}$ is the Hamiltonian density and $Q = E$ is the total energy. For spatial translations, $Q = P_i$ is the momentum. The first theorem produces one conserved charge per continuous symmetry parameter. The key distinction is that a global symmetry with $p$ parameters yields exactly $p$ independent conserved charges, whereas a gauge symmetry (Noether II) yields identities rather than new charges.

Noether's second theorem applies to local (gauge) symmetries parametrised by arbitrary functions. If $S$ is invariant under $\delta_\epsilon \phi = R[\epsilon]$ for arbitrary $\epsilon(x)$, then there exist off-shell identities:

where $R^\dagger$ is the formal adjoint of $R$. For general relativity with diffeomorphism invariance $\delta_\xi g_{\mu\nu} = \nabla_\mu\xi_\nu + \nabla_\nu\xi_\mu$, the identity becomes:

This holds identically, without using the equations of motion. The contracted Bianchi identity is the Noether-II identity for diffeomorphism invariance. It guarantees that the Einstein equations $G_{\mu\nu} = 8\pi G\,T_{\mu\nu}$ automatically imply $\nabla_\mu T^{\mu\nu} = 0$.

For a diffeomorphism generated by $\xi^\mu$, the Wald Noether charge 2-form $Q_\xi$ is constructed from the presymplectic potential:

The integrated charge on a codimension-2 surface $\mathcal{C}$ (a cut of $\mathscr{I}^+$) is:

The Wald–Zoupas correction ensures integrability: the charge variation equals $\delta\mathcal{Q} = \oint_{\mathcal{C}} \iota_\xi\omega$ for the presymplectic current $\omega$. This construction is universal and applies to any diffeomorphism-covariant Lagrangian theory.

For a supertranslation parametrised by $f(x^A)$ on $S^2$, the Wald construction at $\mathscr{I}^+$ yields:

Special cases: for $f = 1$, this gives the total Bondi mass $M_B$. For $f = Y_1^m(\theta,\phi)$ (the $\ell = 1$ spherical harmonics), it gives the three components of Bondi linear momentum:

The $\ell \geq 2$ modes are the genuine supertranslation charges with no Poincare counterpart. The Ward identity for $Q_f$ is equivalent to Weinberg's soft graviton theorem.

For a superrotation generated by a conformal Killing vector $Y^A$ on $S^2$, the complete charge includes the angular momentum aspect $N_A$, the Bondi mass gradient, and a Wald–Zoupas integrability correction:

For the six global Lorentz generators, $Q_Y$ reduces to the ADM angular momentum and boost charges. In Virasoro modes $\ell_n = -z^{n+1}\partial_z$, the charge algebra includes a central extension:

The $\ell \geq 2$ modes of $Y^A$ are the superrotation charges, whose Ward identities give the subleading soft graviton theorem of Cachazo and Strominger.

The following table encodes the complete correspondence between BMS symmetries, their conserved charges, the associated memory effects, and the Perelman analogue under the Ricci flow dictionary:

The DeTurck–Ricci flow modifies the Ricci flow by a Lie derivative term to break diffeomorphism invariance and restore parabolicity:

where $\hat{\Gamma}$ is the connection of a fixed reference metric $\hat{g}$. This flow can be derived from a modified action. Define the DeTurck Lagrangian:

The Noether current associated with the residual symmetry $\xi^i$ of the DeTurck system is:

On a gradient Ricci soliton where $R_{ij} + \nabla_i\nabla_j f = \lambda g_{ij}$, this simplifies to $J_{\mathrm{DT}}^i = \lambda\,\xi^i e^{-f}$, showing that the DeTurck current is proportional to the soliton constant. The conservation law reads:

This is the Bianchi identity for the Perelman system. The DeTurck vector $W^k$ plays the role of a gauge-fixing term analogous to the harmonic gauge condition $\nabla_\mu \bar{h}^{\mu\nu} = 0$ in linearised gravity, and its Noether current encodes the residual gauge structure that survives after fixing.

Each Noether charge satisfies a flux-balance law relating its values on two cuts of $\mathscr{I}^+$ to the flux of radiation between them. For the supertranslation charge:

For $f = 1$, this recovers the Bondi mass-loss formula. For the superrotation charge, the flux involves a more intricate coupling between the news and the Lie derivative of the shear:

The memory effect is the $u_1 \to -\infty$, $u_2 \to +\infty$ limit of these flux-balance laws. The displacement memory measures the total supertranslation flux, while the spin memory measures the total superrotation flux. In the Perelman dictionary, these map to the total entropy production along a finite segment of the Ricci flow.

The two Noether theorems together organise the entire infrared triangle of gravity:

The memory effect completes the triangle: the Ward identity for $Q_\xi$ in scattering amplitudes is the soft theorem, and the classical limit of the soft theorem is the memory effect. Schematically:

Each vertex of this triangle has a Perelman counterpart: BMS symmetry maps to DeTurck gauge symmetry, the soft theorem maps to the dilaton zero mode, and the memory effect maps to the $\mathcal{F}$-functional jump across a Ricci flow singularity. The Ward identity for the soft graviton theorem reads:

This identity states that the S-matrix $\mathcal{S}$ commutes with the supertranslation charge. The classical limit of this Ward identity, applied to coherent gravitational wave states, recovers the displacement memory formula. The entire infrared sector of gravity is thus a consequence of the Noether structure of the Bondi–Sachs action.