Bondi–Sachs Action at Null Infinity

The covariant phase space formalism begins with the presymplectic current. Given a Lagrangian $L[g]$ and a field variation $\delta g$, the symplectic current $\omega^\mu$ is defined as:

where $\Theta^\mu[\delta g]$ is the presymplectic potential obtained from the variation $\delta L = E_{\mu\nu}\delta g^{\mu\nu} + \nabla_\mu \Theta^\mu$. The symplectic form on a Cauchy surface $\Sigma$ is:

We now specialise to Bondi coordinates $(u, r, x^A)$ and take $\Sigma = \Sigma_u$, a constant-$u$ hypersurface extending from $r = r_0$ to $r \to \infty$. The symplectic current $\omega^\mu$ has components that fall off at different rates in $r$. The physically relevant piece is the $r \to \infty$ limit integrated over retarded time and the celestial sphere.

Evaluating $\omega^\mu$ in Bondi gauge, the leading-order contribution involves the news tensor $N_{AB} = \partial_u C_{AB}$ and the shear $C_{AB}$. After careful extraction of the $1/r^2$ piece and integration over the sphere:

This is the Ashtekar–Streubel symplectic form. The conjugate pair on $\mathscr{I}^+$ is $(C_{AB},\, N^{AB})$, analogous to $(q, p)$ in particle mechanics. The shear plays the role of generalised coordinate, and the news is its canonical momentum. This identification is the starting point for constructing the Bondi–Sachs action.

We construct the action whose Euler–Lagrange equations reproduce the Bondi evolution and constraint equations at $\mathscr{I}^+$. The Bondi–Sachs action consists of four terms:

The four terms have distinct physical roles:

• $-m_B$: The Bondi mass aspect, encoding the gravitational energy per unit solid angle. This is the “potential energy” of the system at null infinity.
• $\frac{1}{4}C_{AB}\partial_u N^{AB}$: The kinetic term coupling shear to the time derivative of news. This is the $p\dot{q}$ symplectic term.
• $-\frac{1}{32}N_{AB}N^{AB}$: The news energy density, giving the Bondi news flux. This is the gravitational wave energy radiated per unit retarded time and solid angle.
• $\lambda\,\mathcal{C}$: A Lagrange multiplier enforcing the constraint $\mathcal{C} = 0$, which is the Bondi evolution equation relating $\partial_u m_B$ to the news.

Varying $S_{\mathrm{BS}}$ with respect to $C_{AB}$ (treating $N_{AB} = \partial_u C_{AB}$ as the velocity):

This is precisely the Bondi mass-loss formula. The first term on the right is always non-positive (since $N_{AB}N^{AB} \geq 0$), establishing that the Bondi mass is monotonically decreasing in the absence of incoming radiation. Integrating over $S^2$:

The second derivative term $D^A D^B N_{AB}$ integrates to zero on $S^2$ by the divergence theorem, so the total Bondi mass decreases monotonically. This is the gravitational analogue of the Larmor formula in electrodynamics.

The kinetic term in $S_{\mathrm{BS}}$ involves a time derivative. Integrating by parts in $u$:

The bulk integral produces an additional contribution to the news energy. The crucial new object is the boundary term:

This boundary term is non-zero precisely when gravitational memory is present. A burst of gravitational waves that changes the shear from $C_{AB}^{(-)}$ to $C_{AB}^{(+)}$ while the news is non-zero at the temporal boundaries of $\mathscr{I}^+$ produces a finite boundary contribution. The boundary term measures the total change in the symplectic pairing between shear and news across all of retarded time.

The shear tensor on $S^2$ admits an electric/magnetic (E/B) decomposition via two scalar potentials $\Phi$ and $\Psi$:

The E-mode potential $\Phi$ generates curl-free shear (displacement memory), while the B-mode potential $\Psi$ generates divergence-free shear (spin memory). The boundary term decomposes accordingly:

The B-mode contribution is the spin memory boundary term:

where $\dot{\Psi} = \partial_u \Psi$. This integral is non-zero only when the B-mode shear changes across the full interval of retarded time. The spin memory effect is therefore a boundary observable of the Bondi–Sachs action, directly analogous to how the GHY term captures boundary data of the Einstein–Hilbert action on a spacelike surface.

The Gibbons–Hawking–York boundary term ensures a well-posed variational principle for the Einstein–Hilbert action on a region $\mathcal{M}$ with spacelike boundary $\partial\mathcal{M}$:

The parallel structure is:

Both boundary terms arise from integration by parts of second-derivative terms in their respective actions. The GHY term cancels the $\delta(\partial_n g)$ variation at a spacelike boundary; the spin memory boundary term captures the $\delta(\partial_u C_{AB})$ data at the temporal endpoints of $\mathscr{I}^+$. This parallel reveals that spin memory is the null-boundary analogue of the GHY term.

The constraint $\mathcal{C}$ enforced by the Lagrange multiplier $\lambda$ is the Bondi evolution equation itself. In full, this reads:

Varying $S_{\mathrm{BS}}$ with respect to $\lambda$ enforces $\mathcal{C} = 0$ as a primary constraint. Varying with respect to $m_B$ gives $\partial_u \lambda = 1$, identifying the multiplier with retarded time up to an integration constant. The constraint surface in the phase space $(C_{AB}, N^{AB}, m_B)$ is thus:

The reduced phase space $\Gamma_{\mathrm{phys}}$ inherits the Ashtekar–Streubel symplectic form restricted to solutions of the Bondi evolution equation. This is the arena on which BMS charges act as canonical transformations. The Dirac bracket on the constraint surface coincides with the Poisson bracket of the free data $(C_{AB}, N^{AB})$ precisely because the constraint is first-class with respect to supertranslations.

Perelman's $\mathcal{F}$-functional on a closed manifold $(M, g)$ with dilaton $f$:

Under Ricci flow $\partial_\tau g_{ij} = -2R_{ij}$, the $\mathcal{F}$-functional increases monotonically. Its Euler–Lagrange equations are $R_{ij} + \nabla_i\nabla_j f = 0$, the gradient Ricci soliton equation. The structural parallel with $S_{\mathrm{BS}}$ is:

Under this dictionary, the Bondi mass-loss formula maps to the monotonicity of $\mathcal{F}$:

The spin memory boundary term maps to the change in $\mathcal{F}$ across a finite Ricci flow interval:

Both are gradient-flow dissipation identities controlled by boundary data. The positivity of $d\mathcal{F}/d\tau$ mirrors the negativity of $dM_B/du$, reflecting the irreversible nature of both gravitational radiation and geometric diffusion. This completes the variational bridge between the Bondi–Sachs action at null infinity and Perelman's entropy functional on a Riemannian manifold.